Theory AOT_PLM

1(*<*)
2theory AOT_PLM
3  imports AOT_Axioms
4begin
5(*>*)
6
7section‹The Deductive System PLM›
8
9(* constrain sledgehammer to the abstraction layer *)
10unbundle AOT_no_atp
11
12AOT_theorem "modus-ponens": assumes ‹φ› and ‹φ → ψ› shows ‹ψ›
13  using assms by (simp add: AOT_sem_imp) (* NOTE: semantics needed *)
14lemmas MP = "modus-ponens"
15
16AOT_theorem "non-con-thm-thm": assumes ‹⊢□ φ› shows ‹⊢ φ›
17  using assms by simp
18
19AOT_theorem "vdash-properties:1[1]": assumes ‹φ ∈ Λ› shows ‹⊢ φ›
20  using assms unfolding AOT_model_act_axiom_def by blast (* NOTE: semantics needed *)
21
22text‹Convenience attribute for instantiating modally-fragile axioms.›
23attribute_setup act_axiom_inst =
24  ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[1]"})))›
25  "Instantiate modally fragile axiom as modally fragile theorem."
26
27AOT_theorem "vdash-properties:1[2]": assumes ‹φ ∈ Λ□› shows ‹⊢□ φ›
28  using assms unfolding AOT_model_axiom_def by blast (* NOTE: semantics needed *)
29
30text‹Convenience attribute for instantiating modally-strict axioms.›
31attribute_setup axiom_inst =
32  ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[2]"})))›
33  "Instantiate axiom as theorem."
34
35text‹Convenience methods and theorem sets for applying "cqt:2".›
36method cqt_2_lambda_inst_prover = (fast intro: AOT_instance_of_cqt_2_intro)
37method "cqt:2[lambda]" = (rule "cqt:2[lambda]"[axiom_inst]; cqt_2_lambda_inst_prover)
38lemmas "cqt:2" = "cqt:2[const_var]"[axiom_inst] "cqt:2[lambda]"[axiom_inst] AOT_instance_of_cqt_2_intro
39method "cqt:2" = (safe intro!: "cqt:2")
40
41AOT_theorem "vdash-properties:3": assumes ‹⊢□ φ› shows ‹Γ ⊢ φ›
42  using assms by blast
43
44AOT_theorem "vdash-properties:5": assumes ‹Γ1 ⊢ φ› and ‹Γ2 ⊢ φ → ψ› shows ‹Γ1, Γ2 ⊢ ψ›
45  using MP assms by blast
46
47AOT_theorem "vdash-properties:6": assumes ‹φ› and ‹φ → ψ› shows ‹ψ›
48  using MP assms by blast
49
50AOT_theorem "vdash-properties:8": assumes ‹Γ ⊢ φ› and ‹φ ⊢ ψ› shows ‹Γ ⊢ ψ›
51  using assms by argo
52
53AOT_theorem "vdash-properties:9": assumes ‹φ› shows ‹ψ → φ›
54  using MP "pl:1"[axiom_inst] assms by blast
55
56AOT_theorem "vdash-properties:10": assumes ‹φ → ψ› and ‹φ› shows ‹ψ›
57  using MP assms by blast
58lemmas "→E" = "vdash-properties:10"
59
60AOT_theorem "rule-gen": assumes ‹for arbitrary α: φ{α}› shows ‹∀α φ{α}›
61  using assms by (metis AOT_var_of_term_inverse AOT_sem_denotes AOT_sem_forall) (* NOTE: semantics needed *)
62lemmas GEN = "rule-gen"
63
64AOT_theorem "RN[prem]": assumes ‹Γ ⊢□ φ› shows ‹□Γ ⊢□ □φ›
65  by (meson AOT_sem_box assms image_iff) (* NOTE: semantics needed *)
66AOT_theorem RN: assumes ‹⊢□ φ› shows ‹□φ›
67  using "RN[prem]" assms by blast
68
69AOT_axiom "df-rules-formulas[1]": assumes ‹φ ≡df ψ› shows ‹φ → ψ›
70  using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp) (* NOTE: semantics needed *)
71AOT_axiom "df-rules-formulas[2]": assumes ‹φ ≡df ψ› shows ‹ψ → φ›
72  using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp) (* NOTE: semantics needed *)
73(* NOTE: for convenience also state the above as regular theorems *)
74AOT_theorem "df-rules-formulas[3]": assumes ‹φ ≡df ψ› shows ‹φ → ψ›
75  using "df-rules-formulas[1]"[axiom_inst, OF assms].
76AOT_theorem "df-rules-formulas[4]": assumes ‹φ ≡df ψ› shows ‹ψ → φ›
77  using "df-rules-formulas[2]"[axiom_inst, OF assms].
78
79
80AOT_axiom "df-rules-terms[1]":
81  assumes ‹τ{α1...αn} =df σ{α1...αn}›
82  shows ‹(σ{τ1...τn}↓ → τ{τ1...τn} = σ{τ1...τn}) & (¬σ{τ1...τn}↓ → ¬τ{τ1...τn}↓)›
83  using assms by (simp add: AOT_model_axiomI AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
84AOT_axiom "df-rules-terms[2]":
85  assumes ‹τ =df σ›
86  shows ‹(σ↓ → τ = σ) & (¬σ↓ → ¬τ↓)›
87  by (metis "df-rules-terms[1]" case_unit_Unity assms)
88(* NOTE: for convenience also state the above as regular theorems *)
89AOT_theorem "df-rules-terms[3]":
90  assumes ‹τ{α1...αn} =df σ{α1...αn}›
91  shows ‹(σ{τ1...τn}↓ → τ{τ1...τn} = σ{τ1...τn}) & (¬σ{τ1...τn}↓ → ¬τ{τ1...τn}↓)›
92  using "df-rules-terms[1]"[axiom_inst, OF assms].
93AOT_theorem "df-rules-terms[4]":
94  assumes ‹τ =df σ›
95  shows ‹(σ↓ → τ = σ) & (¬σ↓ → ¬τ↓)›
96  using "df-rules-terms[2]"[axiom_inst, OF assms].
97
98
99AOT_theorem "if-p-then-p": ‹φ → φ›
100  by (meson "pl:1"[axiom_inst] "pl:2"[axiom_inst] MP)
101
102AOT_theorem "deduction-theorem": assumes ‹φ ⊢ ψ› shows ‹φ → ψ›
103  using assms by (simp add: AOT_sem_imp) (* NOTE: semantics needed *)
104lemmas CP = "deduction-theorem"
105lemmas "→I" = "deduction-theorem"
106
107AOT_theorem "ded-thm-cor:1": assumes ‹Γ1 ⊢ φ → ψ› and ‹Γ2 ⊢ ψ → χ› shows ‹Γ1, Γ2 ⊢ φ → χ›
108  using "→E" "→I" assms by blast
109AOT_theorem "ded-thm-cor:2": assumes ‹Γ1 ⊢ φ → (ψ → χ)› and ‹Γ2 ⊢ ψ› shows ‹Γ1, Γ2 ⊢ φ → χ›
110  using "→E" "→I" assms by blast
111
112AOT_theorem "ded-thm-cor:3": assumes ‹φ → ψ› and ‹ψ → χ› shows ‹φ → χ›
113  using "→E" "→I" assms by blast
114declare "ded-thm-cor:3"[trans]
115AOT_theorem "ded-thm-cor:4": assumes ‹φ → (ψ → χ)› and ‹ψ› shows ‹φ → χ›
116  using "→E" "→I" assms by blast
117
118lemmas "Hypothetical Syllogism" = "ded-thm-cor:3"
119
120AOT_theorem "useful-tautologies:1": ‹¬¬φ → φ›
121  by (metis "pl:3"[axiom_inst] "→I" "Hypothetical Syllogism")
122AOT_theorem "useful-tautologies:2": ‹φ → ¬¬φ›
123  by (metis "pl:3"[axiom_inst] "→I" "ded-thm-cor:4")
124AOT_theorem "useful-tautologies:3": ‹¬φ → (φ → ψ)›
125  by (meson "ded-thm-cor:4" "pl:3"[axiom_inst] "→I")
126AOT_theorem "useful-tautologies:4": ‹(¬ψ → ¬φ) → (φ → ψ)›
127  by (meson "pl:3"[axiom_inst] "Hypothetical Syllogism" "→I")
128AOT_theorem "useful-tautologies:5": ‹(φ → ψ) → (¬ψ → ¬φ)›
129  by (metis "useful-tautologies:4" "Hypothetical Syllogism" "→I")
130
131AOT_theorem "useful-tautologies:6": ‹(φ → ¬ψ) → (ψ → ¬φ)›
132  by (metis "→I" MP "useful-tautologies:4")
133
134AOT_theorem "useful-tautologies:7": ‹(¬φ → ψ) → (¬ψ → φ)›
135  by (metis "→I" MP "useful-tautologies:3" "useful-tautologies:5")
136
137AOT_theorem "useful-tautologies:8": ‹φ → (¬ψ → ¬(φ → ψ))›
138  by (metis "→I" MP "useful-tautologies:5")
139
140AOT_theorem "useful-tautologies:9": ‹(φ → ψ) → ((¬φ → ψ) → ψ)›
141  by (metis "→I" MP "useful-tautologies:6")
142
143AOT_theorem "useful-tautologies:10": ‹(φ → ¬ψ) → ((φ → ψ) → ¬φ)›
144  by (metis "→I" MP "pl:3"[axiom_inst])
145
146AOT_theorem "dn-i-e:1": assumes ‹φ› shows ‹¬¬φ›
147  using MP "useful-tautologies:2" assms by blast
148lemmas "¬¬I" = "dn-i-e:1"
149AOT_theorem "dn-i-e:2": assumes ‹¬¬φ› shows ‹φ›
150  using MP "useful-tautologies:1" assms by blast
151lemmas "¬¬E" = "dn-i-e:2"
152
153AOT_theorem "modus-tollens:1": assumes ‹φ → ψ› and ‹¬ψ› shows ‹¬φ›
154  using MP "useful-tautologies:5" assms by blast
155AOT_theorem "modus-tollens:2": assumes ‹φ → ¬ψ› and ‹ψ› shows ‹¬φ›
156  using "¬¬I" "modus-tollens:1" assms by blast
157lemmas MT = "modus-tollens:1" "modus-tollens:2"
158
159AOT_theorem "contraposition:1[1]": assumes ‹φ → ψ› shows ‹¬ψ → ¬φ›
160  using "→I" MT(1) assms by blast
161AOT_theorem "contraposition:1[2]": assumes ‹¬ψ → ¬φ› shows ‹φ → ψ›
162  using "→I" "¬¬E" MT(2) assms by blast
163
164AOT_theorem "contraposition:2": assumes ‹φ → ¬ψ› shows ‹ψ → ¬φ›
165  using "→I" MT(2) assms by blast
166
167(* TODO: this is actually a mixture of the two variants given in PLM; adjust. *)
168AOT_theorem "reductio-aa:1":
169  assumes ‹¬φ ⊢ ¬ψ› and ‹¬φ ⊢ ψ› shows ‹φ›
170  using "→I" "¬¬E" MT(2) assms by blast
171AOT_theorem "reductio-aa:2":
172  assumes ‹φ ⊢ ¬ψ› and ‹φ ⊢ ψ› shows ‹¬φ›
173  using "reductio-aa:1" assms by blast
174lemmas "RAA" = "reductio-aa:1" "reductio-aa:2"
175
176AOT_theorem "exc-mid": ‹φ ∨ ¬φ›
177  using "df-rules-formulas[4]" "if-p-then-p" MP "conventions:2" by blast
178
179AOT_theorem "non-contradiction": ‹¬(φ & ¬φ)›
180  using "df-rules-formulas[3]" MT(2) "useful-tautologies:2" "conventions:1" by blast
181
182AOT_theorem "con-dis-taut:1": ‹(φ & ψ) → φ›
183  by (meson "→I" "df-rules-formulas[3]" MP RAA(1) "conventions:1")
184AOT_theorem "con-dis-taut:2": ‹(φ & ψ) → ψ›
185  by (metis "→I" "df-rules-formulas[3]" MT(2) RAA(2) "¬¬E" "conventions:1")
186lemmas "Conjunction Simplification" = "con-dis-taut:1" "con-dis-taut:2"
187
188AOT_theorem "con-dis-taut:3": ‹φ → (φ ∨ ψ)›
189  by (meson "contraposition:1[2]" "df-rules-formulas[4]" MP "→I" "conventions:2")
190AOT_theorem "con-dis-taut:4": ‹ψ → (φ ∨ ψ)›
191  using "Hypothetical Syllogism" "df-rules-formulas[4]" "pl:1"[axiom_inst] "conventions:2" by blast
192lemmas "Disjunction Addition" = "con-dis-taut:3" "con-dis-taut:4"
193
194AOT_theorem "con-dis-taut:5": ‹φ → (ψ → (φ & ψ))›
195  by (metis "contraposition:2" "Hypothetical Syllogism" "→I" "df-rules-formulas[4]" "conventions:1")
196lemmas Adjunction = "con-dis-taut:5"
197
198AOT_theorem "con-dis-taut:6": ‹(φ & φ) ≡ φ›
199  by (metis Adjunction "→I" "df-rules-formulas[4]" MP "Conjunction Simplification"(1) "conventions:3")
200lemmas "Idempotence of &" = "con-dis-taut:6"
201
202AOT_theorem "con-dis-taut:7": ‹(φ ∨ φ) ≡ φ›
203proof -
204  {
205    AOT_assume ‹φ ∨ φ›
206    AOT_hence ‹¬φ → φ›
207      using "conventions:2"[THEN "df-rules-formulas[3]"] MP by blast
208    AOT_hence ‹φ› using "if-p-then-p" RAA(1) MP by blast
209  }
210  moreover {
211    AOT_assume ‹φ›
212    AOT_hence ‹φ ∨ φ› using "Disjunction Addition"(1) MP by blast
213  }
214  ultimately AOT_show ‹(φ ∨ φ) ≡ φ›
215    using "conventions:3"[THEN "df-rules-formulas[4]"] MP
216    by (metis Adjunction "→I")
217qed
218lemmas "Idempotence of ∨" = "con-dis-taut:7"
219
220
221AOT_theorem "con-dis-i-e:1": assumes ‹φ› and ‹ψ› shows ‹φ & ψ›
222  using Adjunction MP assms by blast
223lemmas "&I" = "con-dis-i-e:1"
224
225AOT_theorem "con-dis-i-e:2:a": assumes ‹φ & ψ› shows ‹φ›
226  using "Conjunction Simplification"(1) MP assms by blast
227AOT_theorem "con-dis-i-e:2:b": assumes ‹φ & ψ› shows ‹ψ›
228  using "Conjunction Simplification"(2) MP assms by blast
229lemmas "&E" = "con-dis-i-e:2:a" "con-dis-i-e:2:b"
230
231AOT_theorem "con-dis-i-e:3:a": assumes ‹φ› shows ‹φ ∨ ψ›
232  using "Disjunction Addition"(1) MP assms by blast
233AOT_theorem "con-dis-i-e:3:b": assumes ‹ψ› shows ‹φ ∨ ψ›
234  using "Disjunction Addition"(2) MP assms by blast
235AOT_theorem "con-dis-i-e:3:c": assumes ‹φ ∨ ψ› and ‹φ → χ› and ‹ψ → Θ› shows ‹χ ∨ Θ›
236  by (metis "con-dis-i-e:3:a" "Disjunction Addition"(2) "df-rules-formulas[3]" MT(1) RAA(1) "conventions:2" assms)
237lemmas "∨I" = "con-dis-i-e:3:a" "con-dis-i-e:3:b" "con-dis-i-e:3:c"
238
239AOT_theorem "con-dis-i-e:4:a": assumes ‹φ ∨ ψ› and ‹φ → χ› and ‹ψ → χ› shows ‹χ›
240  by (metis MP RAA(2) "df-rules-formulas[3]" "conventions:2" assms)
241AOT_theorem "con-dis-i-e:4:b": assumes ‹φ ∨ ψ› and ‹¬φ› shows ‹ψ›
242  using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
243AOT_theorem "con-dis-i-e:4:c": assumes ‹φ ∨ ψ› and ‹¬ψ› shows ‹φ›
244  using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
245lemmas "∨E" = "con-dis-i-e:4:a" "con-dis-i-e:4:b" "con-dis-i-e:4:c"
246
247AOT_theorem "raa-cor:1": assumes ‹¬φ ⊢ ψ & ¬ψ› shows ‹φ›
248  using "&E" "∨E"(3) "∨I"(2) RAA(2) assms by blast
249AOT_theorem "raa-cor:2": assumes ‹φ ⊢ ψ & ¬ψ› shows ‹¬φ›
250  using "raa-cor:1" assms by blast
251AOT_theorem "raa-cor:3": assumes ‹φ› and ‹¬ψ ⊢ ¬φ› shows ‹ψ›
252  using RAA assms by blast
253AOT_theorem "raa-cor:4": assumes ‹¬φ› and ‹¬ψ ⊢ φ› shows ‹ψ›
254  using RAA assms by blast
255AOT_theorem "raa-cor:5": assumes ‹φ› and ‹ψ ⊢ ¬φ› shows ‹¬ψ›
256  using RAA assms by blast
257AOT_theorem "raa-cor:6": assumes ‹¬φ› and ‹ψ ⊢ φ› shows ‹¬ψ›
258  using RAA assms by blast
259
260(* TODO: note these need manual introduction rules *)
261AOT_theorem "oth-class-taut:1:a": ‹(φ → ψ) ≡ ¬(φ & ¬ψ)›
262  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
263     (metis "&E" "&I" "raa-cor:3" "→I" MP)
264AOT_theorem "oth-class-taut:1:b": ‹¬(φ → ψ) ≡ (φ & ¬ψ)›
265  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
266     (metis "&E" "&I" "raa-cor:3" "→I" MP)
267AOT_theorem "oth-class-taut:1:c": ‹(φ → ψ) ≡ (¬φ ∨ ψ)›
268  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
269     (metis "&I" "∨I"(1, 2) "∨E"(3) "→I" MP "raa-cor:1")
270
271AOT_theorem "oth-class-taut:2:a": ‹(φ & ψ) ≡ (ψ & φ)›
272  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
273     (meson "&I" "&E" "→I")
274lemmas "Commutativity of &" = "oth-class-taut:2:a"
275AOT_theorem "oth-class-taut:2:b": ‹(φ & (ψ & χ)) ≡ ((φ & ψ) & χ)›
276  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
277     (metis "&I" "&E" "→I")
278lemmas "Associativity of &" = "oth-class-taut:2:b"
279AOT_theorem "oth-class-taut:2:c": ‹(φ ∨ ψ) ≡ (ψ ∨ φ)›
280  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
281     (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
282lemmas "Commutativity of ∨" = "oth-class-taut:2:c"
283AOT_theorem "oth-class-taut:2:d": ‹(φ ∨ (ψ ∨ χ)) ≡ ((φ ∨ ψ) ∨ χ)›
284  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
285     (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
286lemmas "Associativity of ∨" = "oth-class-taut:2:d"
287AOT_theorem "oth-class-taut:2:e": ‹(φ ≡ ψ) ≡ (ψ ≡ φ)›
288  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I";
289      metis "&I" "df-rules-formulas[4]" "conventions:3" "&E" "Hypothetical Syllogism" "→I" "df-rules-formulas[3]")
290lemmas "Commutativity of ≡" = "oth-class-taut:2:e"
291AOT_theorem "oth-class-taut:2:f": ‹(φ ≡ (ψ ≡ χ)) ≡ ((φ ≡ ψ) ≡ χ)›
292  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
293        "→I" "→E" "&E" "&I"
294  by metis
295lemmas "Associativity of ≡" = "oth-class-taut:2:f"
296
297AOT_theorem "oth-class-taut:3:a": ‹φ ≡ φ›
298  using "&I" "vdash-properties:6" "if-p-then-p" "df-rules-formulas[4]" "conventions:3" by blast
299AOT_theorem "oth-class-taut:3:b": ‹φ ≡ ¬¬φ›
300  using "&I" "useful-tautologies:1" "useful-tautologies:2" "vdash-properties:6" "df-rules-formulas[4]" "conventions:3" by blast
301AOT_theorem "oth-class-taut:3:c": ‹¬(φ ≡ ¬φ)›
302  by (metis "&E" "→E" RAA "df-rules-formulas[3]" "conventions:3")
303
304AOT_theorem "oth-class-taut:4:a": ‹(φ → ψ) → ((ψ → χ) → (φ → χ))›
305  by (metis "→E" "→I")
306AOT_theorem "oth-class-taut:4:b": ‹(φ ≡ ψ) ≡ (¬φ ≡ ¬ψ)›
307  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
308        "→I" "→E" "&E" "&I" RAA by metis
309AOT_theorem "oth-class-taut:4:c": ‹(φ ≡ ψ) → ((φ → χ) ≡ (ψ → χ))›
310  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
311        "→I" "→E" "&E" "&I" by metis
312AOT_theorem "oth-class-taut:4:d": ‹(φ ≡ ψ) → ((χ → φ) ≡ (χ → ψ))›
313  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
314        "→I" "→E" "&E" "&I" by metis
315AOT_theorem "oth-class-taut:4:e": ‹(φ ≡ ψ) → ((φ & χ) ≡ (ψ & χ))›
316  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
317        "→I" "→E" "&E" "&I" by metis
318AOT_theorem "oth-class-taut:4:f": ‹(φ ≡ ψ) → ((χ & φ) ≡ (χ & ψ))›
319  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
320        "→I" "→E" "&E" "&I" by metis
321AOT_theorem "oth-class-taut:4:g": ‹(φ ≡ ψ) ≡ ((φ & ψ) ∨ (¬φ & ¬ψ))›
322proof(safe intro!: "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"] "&I" "→I"
323           dest!: "conventions:3"[THEN "df-rules-formulas[3]", THEN "→E"])
324  AOT_show ‹φ & ψ ∨ (¬φ & ¬ψ)› if ‹(φ → ψ) & (ψ → φ)›
325    using "&E" "∨I" "→E" "&I" "raa-cor:1" "→I" "∨E" that by metis
326next
327  AOT_show ‹ψ› if ‹φ & ψ ∨ (¬φ & ¬ψ)› and ‹φ›
328    using that "∨E" "&E" "raa-cor:3" by blast
329next
330  AOT_show ‹φ› if ‹φ & ψ ∨ (¬φ & ¬ψ)› and ‹ψ›
331    using that "∨E" "&E" "raa-cor:3" by blast
332qed
333AOT_theorem "oth-class-taut:4:h": ‹¬(φ ≡ ψ) ≡ ((φ & ¬ψ) ∨ (¬φ & ψ))›
334proof (safe intro!: "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"] "&I" "→I")
335  AOT_show ‹φ & ¬ψ ∨ (¬φ & ψ)› if ‹¬(φ ≡ ψ)›
336    by (metis that "&I" "∨I"(1, 2) "→I" MT(1) "df-rules-formulas[4]" "raa-cor:3" "conventions:3")
337next
338  AOT_show ‹¬(φ ≡ ψ)› if ‹φ & ¬ψ ∨ (¬φ & ψ)›
339    by (metis that "&E" "∨E"(2) "→E" "df-rules-formulas[3]" "raa-cor:3" "conventions:3")
340qed
341AOT_theorem "oth-class-taut:5:a": ‹(φ & ψ) ≡ ¬(¬φ ∨ ¬ψ)›
342  using "conventions:3"[THEN "df-rules-formulas[4]"]
343        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
344AOT_theorem "oth-class-taut:5:b": ‹(φ ∨ ψ) ≡ ¬(¬φ & ¬ψ)›
345  using "conventions:3"[THEN "df-rules-formulas[4]"]
346        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
347AOT_theorem "oth-class-taut:5:c": ‹¬(φ & ψ) ≡ (¬φ ∨ ¬ψ)›
348  using "conventions:3"[THEN "df-rules-formulas[4]"]
349        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
350AOT_theorem "oth-class-taut:5:d": ‹¬(φ ∨ ψ) ≡ (¬φ & ¬ψ)›
351  using "conventions:3"[THEN "df-rules-formulas[4]"]
352        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
353
354lemmas DeMorgan = "oth-class-taut:5:c" "oth-class-taut:5:d"
355
356AOT_theorem "oth-class-taut:6:a": ‹(φ & (ψ ∨ χ)) ≡ ((φ & ψ) ∨ (φ & χ))›
357  using "conventions:3"[THEN "df-rules-formulas[4]"]
358        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
359AOT_theorem "oth-class-taut:6:b": ‹(φ ∨ (ψ & χ)) ≡ ((φ ∨ ψ) & (φ ∨ χ))›
360  using "conventions:3"[THEN "df-rules-formulas[4]"]
361        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
362
363AOT_theorem "oth-class-taut:7:a": ‹((φ & ψ) → χ) → (φ → (ψ → χ))›
364  by (metis "&I" "→E" "→I")
365lemmas Exportation = "oth-class-taut:7:a"
366AOT_theorem "oth-class-taut:7:b": ‹(φ → (ψ →χ)) → ((φ & ψ) → χ)›
367  by (metis "&E" "→E" "→I")
368lemmas Importation = "oth-class-taut:7:b"
369
370AOT_theorem "oth-class-taut:8:a": ‹(φ → (ψ → χ)) ≡ (ψ → (φ → χ))›
371  using "conventions:3"[THEN "df-rules-formulas[4]"] "→I" "→E" "&E" "&I" by metis
372lemmas Permutation = "oth-class-taut:8:a"
373AOT_theorem "oth-class-taut:8:b": ‹(φ → ψ) → ((φ → χ) → (φ → (ψ & χ)))›
374  by (metis "&I" "→E" "→I")
375lemmas Composition = "oth-class-taut:8:b"
376AOT_theorem "oth-class-taut:8:c": ‹(φ → χ) → ((ψ → χ) → ((φ ∨ ψ) → χ))›
377  by (metis "∨E"(2) "→E" "→I" RAA(1))
378AOT_theorem "oth-class-taut:8:d": ‹((φ → ψ) & (χ → Θ)) → ((φ & χ) → (ψ & Θ))›
379  by (metis "&E" "&I" "→E" "→I")
380lemmas "Double Composition" = "oth-class-taut:8:d"
381AOT_theorem "oth-class-taut:8:e": ‹((φ & ψ) ≡ (φ & χ)) ≡ (φ → (ψ ≡ χ))›
382  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
383            "→I" "→E" "&E" "&I")
384AOT_theorem "oth-class-taut:8:f": ‹((φ & ψ) ≡ (χ & ψ)) ≡ (ψ → (φ ≡ χ))›
385  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
386            "→I" "→E" "&E" "&I")
387AOT_theorem "oth-class-taut:8:g": ‹(ψ ≡ χ) → ((φ ∨ ψ) ≡ (φ ∨ χ))›
388  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
389            "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
390AOT_theorem "oth-class-taut:8:h": ‹(ψ ≡ χ) → ((ψ ∨ φ) ≡ (χ ∨ φ))›
391  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
392            "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
393AOT_theorem "oth-class-taut:8:i": ‹(φ ≡ (ψ & χ)) → (ψ → (φ ≡ χ))›
394  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
395            "→I" "→E" "&E" "&I")
396
397AOT_theorem "intro-elim:1": assumes ‹φ ∨ ψ› and ‹φ ≡ χ› and ‹ψ ≡ Θ› shows ‹χ ∨ Θ›
398  by (metis assms "∨I"(1, 2) "∨E"(1) "conventions:3"[THEN "df-rules-formulas[3]"] "→I" "→E" "&E"(1))
399
400AOT_theorem "intro-elim:2": assumes ‹φ → ψ› and ‹ψ → φ› shows ‹φ ≡ ψ›
401  by (meson "&I" "conventions:3" "df-rules-formulas[4]" MP assms)
402lemmas "≡I" = "intro-elim:2"
403
404AOT_theorem "intro-elim:3:a": assumes ‹φ ≡ ψ› and ‹φ› shows ‹ψ›
405  by (metis "∨I"(1) "→I" "∨E"(1) "intro-elim:1" assms)
406AOT_theorem "intro-elim:3:b": assumes ‹φ ≡ ψ› and ‹ψ› shows ‹φ›
407  using "intro-elim:3:a" "Commutativity of ≡" assms by blast
408AOT_theorem "intro-elim:3:c": assumes ‹φ ≡ ψ› and ‹¬φ› shows ‹¬ψ›
409  using "intro-elim:3:b" "raa-cor:3" assms by blast
410AOT_theorem "intro-elim:3:d": assumes ‹φ ≡ ψ› and ‹¬ψ› shows ‹¬φ›
411  using "intro-elim:3:a" "raa-cor:3" assms by blast
412AOT_theorem "intro-elim:3:e": assumes ‹φ ≡ ψ› and ‹ψ ≡ χ› shows ‹φ ≡ χ›
413  by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
414declare "intro-elim:3:e"[trans]
415AOT_theorem "intro-elim:3:f": assumes ‹φ ≡ ψ› and ‹φ ≡ χ› shows ‹χ ≡ ψ›
416  by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
417lemmas "≡E" = "intro-elim:3:a" "intro-elim:3:b" "intro-elim:3:c" "intro-elim:3:d" "intro-elim:3:e" "intro-elim:3:f"
418
419declare "Commutativity of ≡"[THEN "≡E"(1), sym]
420
421AOT_theorem "rule-eq-df:1": assumes ‹φ ≡df ψ› shows ‹φ ≡ ψ›
422  by (simp add: "≡I" "df-rules-formulas[3]" "df-rules-formulas[4]" assms)
423lemmas "≡Df" = "rule-eq-df:1"
424AOT_theorem "rule-eq-df:2": assumes ‹φ ≡df ψ› and ‹φ› shows ‹ψ›
425  using "≡Df" "≡E"(1) assms by blast
426lemmas "≡dfE" = "rule-eq-df:2"
427AOT_theorem "rule-eq-df:3": assumes ‹φ ≡df ψ› and ‹ψ› shows ‹φ›
428  using "≡Df" "≡E"(2) assms by blast
429lemmas "≡dfI" = "rule-eq-df:3"
430
431AOT_theorem  "df-simplify:1": assumes ‹φ ≡ (ψ & χ)› and ‹ψ› shows ‹φ ≡ χ›
432  by (metis "&E"(2) "&I" "≡E"(1, 2) "≡I" "→I" assms)
433(* TODO: this is a slight variation from PLM *)
434AOT_theorem  "df-simplify:2": assumes ‹φ ≡ (ψ & χ)› and ‹χ› shows ‹φ ≡ ψ›
435  by (metis "&E"(1) "&I" "≡E"(1, 2) "≡I" "→I" assms)
436lemmas "≡S" = "df-simplify:1"  "df-simplify:2"
437
438AOT_theorem "rule-ui:1": assumes ‹∀α φ{α}› and ‹τ↓› shows ‹φ{τ}›
439  using "→E" "cqt:1"[axiom_inst] assms by blast
440AOT_theorem "rule-ui:2[const_var]": assumes ‹∀α φ{α}› shows ‹φ{β}›
441  by (simp add: "rule-ui:1" "cqt:2[const_var]"[axiom_inst] assms)
442(* TODO: precise proviso in PLM *)
443AOT_theorem "rule-ui:2[lambda]":
444  assumes ‹∀F φ{F}› and ‹INSTANCE_OF_CQT_2(ψ)›
445  shows ‹φ{[λν1...νn ψ{ν1...νn}]}›
446  by (simp add: "rule-ui:1" "cqt:2[lambda]"[axiom_inst] assms)
447AOT_theorem "rule-ui:3": assumes ‹∀α φ{α}› shows ‹φ{α}›
448  by (simp add: "rule-ui:2[const_var]" assms)
449lemmas "∀E" = "rule-ui:1" "rule-ui:2[const_var]" "rule-ui:2[lambda]" "rule-ui:3"
450
451AOT_theorem "cqt-orig:1[const_var]": ‹∀α φ{α} → φ{β}› by (simp add: "∀E"(2) "→I")
452AOT_theorem "cqt-orig:1[lambda]":
453  assumes ‹INSTANCE_OF_CQT_2(ψ)›
454  shows ‹∀F φ{F} → φ{[λν1...νn ψ{ν1...νn}]}›
455  by (simp add: "∀E"(3) "→I" assms)
456AOT_theorem "cqt-orig:2": ‹∀α (φ → ψ{α}) → (φ → ∀α ψ{α})›
457  by (metis "→I" GEN "vdash-properties:6" "∀E"(4))
458AOT_theorem "cqt-orig:3": ‹∀α φ{α} → φ{α}› using "cqt-orig:1[const_var]" .
459
460(* TODO: work out difference to GEN *)
461AOT_theorem universal: assumes ‹for arbitrary β: φ{β}› shows ‹∀α φ{α}›
462  using GEN assms .
463lemmas "∀I" = universal
464
465(* Generalized mechanism for "∀I" followed by ∀E *)
466ML‹
467fun get_instantiated_allI ctxt varname thm = let
468val trm = Thm.concl_of thm
469val trm = case trm of (@{const Trueprop} $ (@{const AOT_model_valid_in} $ _ $ x)) => x
470                      | _ => raise Term.TERM ("Expected simple theorem.", [trm])
471fun extractVars (Const (const_name‹AOT_term_of_var›, _) $ Var v) =
472    (if fst (fst v) = fst varname then [Var v] else []) (* TODO: care about the index? *)
473  | extractVars (t1 $ t2) = extractVars t1 @ extractVars t2
474  | extractVars (Abs (_, _, t)) = extractVars t
475  | extractVars _ = []
476val vars = extractVars trm
477val vars = fold Term.add_vars vars []
478val var = hd vars
479val trmty = case (snd var) of (Type (type_name‹AOT_var›, [t])) => (t)
480              | _ => raise Term.TYPE ("Expected variable type.", [snd var], [Var var])
481val trm = Abs (Term.string_of_vname (fst var), trmty, Term.abstract_over (
482      Const (const_name‹AOT_term_of_var›, Type ("fun", [snd var, trmty]))
483       $ Var var, trm))
484val trm = Thm.cterm_of (Context.proof_of ctxt) trm
485val ty = hd (Term.add_tvars (Thm.prop_of @{thm "∀I"}) [])
486val typ = Thm.ctyp_of (Context.proof_of ctxt) trmty
487val allthm = Drule.instantiate_normalize ([(ty, typ)],[]) @{thm "∀I"}
488val phi = hd (Term.add_vars (Thm.prop_of allthm) [])
489val allthm = Drule.instantiate_normalize ([],[(phi,trm)]) allthm
490in
491allthm
492end
493›
494
495attribute_setup "∀I" =
496  ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
497  (fn ctxt => fn thm => fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm))›
498  "Quantify over a variable in a theorem using GEN."
499
500attribute_setup "unvarify" =
501  ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
502  (fn ctxt => fn thm =>
503    let
504    val thm = fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm
505    val thm = fold (fn _ => fn thm => thm RS @{thm "∀E"(1)}) args thm
506    in
507     thm
508    end))›
509  "Generalize a statement about variables to a statement about denoting terms."
510
511(* TODO: rereplace-lem does not apply to the embedding *)
512
513AOT_theorem "cqt-basic:1": ‹∀α∀β φ{α,β} ≡ ∀β∀α φ{α,β}›
514  by (metis "≡I" "∀E"(2) "∀I" "→I")
515
516AOT_theorem "cqt-basic:2": ‹∀α(φ{α} ≡ ψ{α}) ≡ (∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α}))›
517proof (rule "≡I"; rule "→I")
518  AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
519  AOT_hence ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
520  AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
521    using "≡E"(1,2) "→I" by blast+
522  AOT_thus ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
523    by (auto intro: "&I" "∀I")
524next
525  AOT_assume ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
526  AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
527    using "∀E"(2) "&E" by blast+
528  AOT_hence ‹φ{α} ≡ ψ{α}› for α
529    using "≡I" by blast
530  AOT_thus ‹∀α(φ{α} ≡ ψ{α})› by (auto intro: "∀I")
531qed
532
533AOT_theorem "cqt-basic:3": ‹∀α(φ{α} ≡ ψ{α}) → (∀α φ{α} ≡ ∀α ψ{α})›
534proof(rule "→I")
535  AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
536  AOT_hence 1: ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
537  {
538    AOT_assume ‹∀α φ{α}›
539    AOT_hence ‹∀α ψ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
540  }
541  moreover {
542    AOT_assume ‹∀α ψ{α}›
543    AOT_hence ‹∀α φ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
544  }
545  ultimately AOT_show ‹∀α φ{α} ≡ ∀α ψ{α}›
546    using "≡I" "→I" by auto
547qed
548
549AOT_theorem "cqt-basic:4": ‹∀α(φ{α} & ψ{α}) → (∀α φ{α} & ∀α ψ{α})›
550proof(rule "→I")
551  AOT_assume 0: ‹∀α(φ{α} & ψ{α})›
552  AOT_have ‹φ{α}› and ‹ψ{α}› for α using "∀E"(2) 0 "&E" by blast+
553  AOT_thus ‹∀α φ{α} & ∀α ψ{α}›
554    by (auto intro: "∀I" "&I")
555qed
556
557AOT_theorem "cqt-basic:5": ‹(∀α1...∀αn(φ{α1...αn})) → φ{α1...αn}›
558  using "cqt-orig:3" by blast
559
560AOT_theorem "cqt-basic:6": ‹∀α∀α φ{α} ≡ ∀α φ{α}›
561  by (meson "≡I" "→I" GEN "cqt-orig:1[const_var]")
562
563AOT_theorem "cqt-basic:7": ‹(φ → ∀α ψ{α}) ≡ ∀α(φ → ψ{α})›
564  by (metis "→I" "vdash-properties:6" "rule-ui:3" "≡I" GEN)
565
566AOT_theorem "cqt-basic:8": ‹(∀α φ{α} ∨ ∀α ψ{α}) → ∀α (φ{α} ∨ ψ{α})›
567  by (simp add: "∨I"(3) "→I" GEN "cqt-orig:1[const_var]")
568
569AOT_theorem "cqt-basic:9": ‹(∀α (φ{α} → ψ{α}) & ∀α (ψ{α} → χ{α})) → ∀α(φ{α} → χ{α})›
570proof -
571  {
572    AOT_assume ‹∀α (φ{α} → ψ{α})›
573    moreover AOT_assume ‹∀α (ψ{α} → χ{α})›
574    ultimately AOT_have ‹φ{α} → ψ{α}› and ‹ψ{α} → χ{α}› for α using "∀E" by blast+
575    AOT_hence ‹φ{α} → χ{α}› for α by (metis "→E" "→I")
576    AOT_hence ‹∀α(φ{α} → χ{α})› using "∀I" by fast
577  }
578  thus ?thesis using "&I" "→I" "&E" by meson
579qed
580
581AOT_theorem "cqt-basic:10": ‹(∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})) → ∀α (φ{α} ≡ χ{α})›
582proof(rule "→I"; rule "∀I")
583  fix β
584  AOT_assume ‹∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})›
585  AOT_hence ‹φ{β} ≡ ψ{β}› and ‹ψ{β} ≡ χ{β}› using "&E" "∀E" by blast+
586  AOT_thus ‹φ{β} ≡ χ{β}› using "≡I" "≡E" by blast
587qed
588
589AOT_theorem "cqt-basic:11": ‹∀α(φ{α} ≡ ψ{α}) ≡ ∀α (ψ{α} ≡ φ{α})›
590proof (rule "≡I"; rule "→I")
591  AOT_assume 0: ‹∀α(φ{α} ≡ ψ{α})›
592  {
593    fix α
594    AOT_have ‹φ{α} ≡ ψ{α}› using 0 "∀E" by blast
595    AOT_hence ‹ψ{α} ≡ φ{α}› using "≡I" "≡E" "→I" "→E" by metis
596  }
597  AOT_thus ‹∀α(ψ{α} ≡ φ{α})› using "∀I" by fast
598next
599  AOT_assume 0: ‹∀α(ψ{α} ≡ φ{α})›
600  {
601    fix α
602    AOT_have ‹ψ{α} ≡ φ{α}› using 0 "∀E" by blast
603    AOT_hence ‹φ{α} ≡ ψ{α}› using "≡I" "≡E" "→I" "→E" by metis
604  }
605  AOT_thus ‹∀α(φ{α} ≡ ψ{α})› using "∀I" by fast
606qed
607
608AOT_theorem "cqt-basic:12": ‹∀α φ{α} → ∀α (ψ{α} → φ{α})›
609  by (simp add: "∀E"(2) "→I" GEN)
610
611AOT_theorem "cqt-basic:13": ‹∀α φ{α} ≡ ∀β φ{β}›
612  using "≡I" "→I" by blast
613
614AOT_theorem "cqt-basic:14": ‹(∀α1...∀αn (φ{α1...αn} → ψ{α1...αn})) → ((∀α1...∀αn φ{α1...αn}) → (∀α1...∀αn ψ{α1...αn}))›
615  using "cqt:3"[axiom_inst] by auto
616
617AOT_theorem "cqt-basic:15": ‹(∀α1...∀αn (φ → ψ{α1...αn})) → (φ → (∀α1...∀αn ψ{α1...αn}))›
618  using "cqt-orig:2" by auto
619
620(* TODO: once more the same in the embedding... need to distinguish these better *)
621AOT_theorem "universal-cor": assumes ‹for arbitrary β: φ{β}›  shows ‹∀α φ{α}›
622  using GEN assms .
623
624AOT_theorem "existential:1": assumes ‹φ{τ}› and ‹τ↓› shows ‹∃α φ{α}›
625proof(rule "raa-cor:1")
626  AOT_assume ‹¬∃α φ{α}›
627  AOT_hence ‹∀α ¬φ{α}›
628    using "≡dfI" "conventions:4" RAA "&I" by blast
629  AOT_hence ‹¬φ{τ}› using assms(2) "∀E"(1) "→E" by blast
630  AOT_thus ‹φ{τ} & ¬φ{τ}› using assms(1) "&I" by blast
631qed
632
633AOT_theorem "existential:2[const_var]": assumes ‹φ{β}› shows ‹∃α φ{α}›
634  using "existential:1" "cqt:2[const_var]"[axiom_inst] assms by blast
635
636AOT_theorem "existential:2[lambda]":
637  assumes ‹φ{[λν1...νn ψ{ν1...νn}]}› and ‹INSTANCE_OF_CQT_2(ψ)›
638  shows ‹∃α φ{α}›
639  using "existential:1" "cqt:2[lambda]"[axiom_inst] assms by blast
640lemmas "∃I" = "existential:1" "existential:2[const_var]" "existential:2[lambda]" 
641
642AOT_theorem "instantiation":
643  assumes ‹for arbitrary β: φ{β} ⊢ ψ› and ‹∃α φ{α}›
644  shows ‹ψ›
645  by (metis (no_types, lifting) "≡dfE" GEN "raa-cor:3" "conventions:4" assms)
646lemmas "∃E" = "instantiation"
647
648AOT_theorem "cqt-further:1": ‹∀α φ{α} → ∃α φ{α}›
649  using "∀E"(4) "∃I"(2) "→I" by metis
650
651AOT_theorem "cqt-further:2": ‹¬∀α φ{α} → ∃α ¬φ{α}›
652  using "∀I" "∃I"(2) "→I" RAA by metis
653
654AOT_theorem "cqt-further:3": ‹∀α φ{α} ≡ ¬∃α ¬φ{α}›
655  using "∀E"(4) "∃E" "→I" RAA
656  by (metis "cqt-further:2" "≡I" "modus-tollens:1")
657
658AOT_theorem "cqt-further:4": ‹¬∃α φ{α} → ∀α ¬φ{α}›
659  using "∀I" "∃I"(2)"→I" RAA by metis
660
661AOT_theorem "cqt-further:5": ‹∃α (φ{α} & ψ{α}) → (∃α φ{α} & ∃α ψ{α})›
662  by (metis (no_types, lifting) "&E" "&I" "∃E" "∃I"(2) "→I")
663
664AOT_theorem "cqt-further:6": ‹∃α (φ{α} ∨ ψ{α}) → (∃α φ{α} ∨ ∃α ψ{α})›
665  by (metis (mono_tags, lifting) "∃E" "∃I"(2) "∨E"(3) "∨I"(1, 2) "→I" RAA(2))
666
667AOT_theorem "cqt-further:7": ‹∃α φ{α} ≡ ∃β φ{β}› (* TODO: vacuous in the embedding *)
668  by (simp add: "oth-class-taut:3:a")
669
670AOT_theorem "cqt-further:8": ‹(∀α φ{α} & ∀α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
671  by (metis (mono_tags, lifting) "&E" "≡I" "∀E"(2) "→I" GEN)
672
673AOT_theorem "cqt-further:9": ‹(¬∃α φ{α} & ¬∃α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
674  by (metis (mono_tags, lifting) "&E" "≡I" "∃I"(2) "→I" GEN "raa-cor:4")
675
676AOT_theorem "cqt-further:10": ‹(∃α φ{α} & ¬∃α ψ{α}) → ¬∀α (φ{α} ≡ ψ{α})›
677proof(rule "→I"; rule "raa-cor:2")
678  AOT_assume 0: ‹∃α φ{α} & ¬∃α ψ{α}›
679  then AOT_obtain α where ‹φ{α}› using "∃E" "&E"(1) by metis
680  moreover AOT_assume ‹∀α (φ{α} ≡ ψ{α})›
681  ultimately AOT_have ‹ψ{α}› using "∀E"(4) "≡E"(1) by blast
682  AOT_hence ‹∃α ψ{α}› using "∃I" by blast
683  AOT_thus ‹∃α ψ{α} & ¬∃α ψ{α}› using 0 "&E"(2) "&I" by blast
684qed
685
686AOT_theorem "cqt-further:11": ‹∃α∃β φ{α,β} ≡ ∃β∃α φ{α,β}›
687  using "≡I" "→I" "∃I"(2) "∃E" by metis
688
689AOT_theorem "log-prop-prop:1": ‹[λ φ]↓›
690  using "cqt:2[lambda0]"[axiom_inst] by auto
691
692AOT_theorem "log-prop-prop:2": ‹φ↓›
693  by (rule "≡dfI"[OF "existence:3"]) "cqt:2[lambda]"
694
695AOT_theorem "exist-nec": ‹τ↓ → □τ↓›
696proof -
697  AOT_have ‹∀β □β↓›
698    by (simp add: GEN RN "cqt:2[const_var]"[axiom_inst])
699  AOT_thus ‹τ↓ → □τ↓›
700    using "cqt:1"[axiom_inst] "→E" by blast
701qed
702
703(* TODO: replace this mechanism by a "proof by types" command *)
704class AOT_Term_id = AOT_Term +
705  assumes "t=t-proper:1"[AOT]: ‹[v ⊨ τ = τ' → τ↓]›
706      and "t=t-proper:2"[AOT]: ‹[v ⊨ τ = τ' → τ'↓]›
707
708instance κ :: AOT_Term_id
709proof
710  AOT_modally_strict {
711    AOT_show ‹κ = κ' → κ↓› for κ κ'
712    proof(rule "→I")
713      AOT_assume ‹κ = κ'›
714      AOT_hence ‹O!κ ∨ A!κ›
715        by (rule "∨I"(3)[OF "≡dfE"[OF "identity:1"]])
716           (meson "→I" "∨I"(1) "&E"(1))+
717      AOT_thus ‹κ↓›
718        by (rule "∨E"(1))
719           (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
720    qed
721  }
722next
723  AOT_modally_strict {
724    AOT_show ‹κ = κ' → κ'↓› for κ κ'
725    proof(rule "→I")
726      AOT_assume ‹κ = κ'›
727      AOT_hence ‹O!κ' ∨ A!κ'›
728        by (rule "∨I"(3)[OF "≡dfE"[OF "identity:1"]])
729           (meson "→I" "∨I" "&E")+
730      AOT_thus ‹κ'↓›
731        by (rule "∨E"(1))
732           (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
733    qed
734  }
735qed
736
737instance rel :: (AOT_κs) AOT_Term_id
738proof
739  AOT_modally_strict {
740    AOT_show ‹Π = Π' → Π↓› for Π Π' :: ‹<'a>› (* TODO: how to get rid of the fixes? *)
741    proof(rule "→I")
742      AOT_assume ‹Π = Π'›
743      AOT_thus ‹Π↓› using "≡dfE"[OF "identity:3"[of Π Π']] "&E" by blast
744    qed
745  }
746next
747  AOT_modally_strict {
748    AOT_show ‹Π = Π' → Π'↓› for Π Π' :: ‹<'a>› (* TODO: how to get rid of the fixes? *)
749    proof(rule "→I")
750      AOT_assume ‹Π = Π'›
751      AOT_thus ‹Π'↓› using "≡dfE"[OF "identity:3"[of Π Π']] "&E" by blast
752    qed
753  }
754qed
755
756instance 𝗈 :: AOT_Term_id
757proof
758  AOT_modally_strict {
759    fix φ ψ
760    AOT_show ‹φ = ψ → φ↓›
761    proof(rule "→I")
762      AOT_assume ‹φ = ψ›
763      AOT_thus ‹φ↓› using "≡dfE"[OF "identity:4"[of φ ψ]] "&E" by blast
764    qed
765  }
766next
767  AOT_modally_strict {
768    fix φ ψ
769    AOT_show ‹φ = ψ → ψ↓›
770    proof(rule "→I")
771      AOT_assume ‹φ = ψ›
772      AOT_thus ‹ψ↓› using "≡dfE"[OF "identity:4"[of φ ψ]] "&E" by blast
773    qed
774  }
775qed
776
777instance prod :: (AOT_Term_id, AOT_Term_id) AOT_Term_id
778proof
779  AOT_modally_strict {
780    fix τ τ' :: ‹'a×'b›
781    AOT_show ‹τ = τ' → τ↓›
782    proof (induct τ; induct τ'; rule "→I")
783      fix τ1 τ1' :: 'a and τ2  τ2' :: 'b
784      AOT_assume ‹«(τ1, τ2)» = «(τ1', τ2')»›
785      AOT_hence ‹(τ1 = τ1') & (τ2 = τ2')› by (metis "≡dfE" tuple_identity_1)
786      AOT_hence ‹τ1↓› and ‹τ2↓› using "t=t-proper:1" "&E" "vdash-properties:10" by blast+
787      AOT_thus ‹«(τ1, τ2)»↓› by (metis "≡dfI" "&I" tuple_denotes)
788    qed
789  }
790next
791  AOT_modally_strict {
792    fix τ τ' :: ‹'a×'b›
793    AOT_show ‹τ = τ' → τ'↓›
794    proof (induct τ; induct τ'; rule "→I")
795      fix τ1 τ1' :: 'a and τ2  τ2' :: 'b
796      AOT_assume ‹«(τ1, τ2)» = «(τ1', τ2')»›
797      AOT_hence ‹(τ1 = τ1') & (τ2 = τ2')› by (metis "≡dfE" tuple_identity_1)
798      AOT_hence ‹τ1'↓› and ‹τ2'↓› using "t=t-proper:2" "&E" "vdash-properties:10" by blast+
799      AOT_thus ‹«(τ1', τ2')»↓› by (metis "≡dfI" "&I" tuple_denotes)
800    qed
801  }
802qed
803
804(* TODO: this is the end of the "proof by types" and makes the results available on new theorems *)
805AOT_register_type_constraints
806  Term: ‹_::AOT_Term_id› ‹_::AOT_Term_id›
807AOT_register_type_constraints
808  Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id}›
809AOT_register_type_constraints
810  Relation: ‹<_::{AOT_κs, AOT_Term_id}>›
811
812AOT_theorem "id-rel-nec-equiv:1": ‹Π = Π' → □∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn)›
813proof(rule "→I")
814  AOT_assume assumption: ‹Π = Π'›
815  AOT_hence ‹Π↓› and ‹Π'↓›
816    using "t=t-proper:1" "t=t-proper:2" MP by blast+
817  moreover AOT_have ‹∀F∀G (F = G → ((□∀x1...∀xn ([F]x1...xn ≡ [F]x1...xn)) → □∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)))›
818    apply (rule GEN)+ using "l-identity"[axiom_inst] by force
819  ultimately AOT_have ‹Π = Π' → ((□∀x1...∀xn ([Π]x1...xn ≡ [Π]x1...xn)) → □∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn))›
820    using "∀E"(1) by blast
821  AOT_hence ‹(□∀x1...∀xn ([Π]x1...xn ≡ [Π]x1...xn)) → □∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn)›
822    using assumption "→E" by blast
823  moreover AOT_have ‹□∀x1...∀xn ([Π]x1...xn ≡ [Π]x1...xn)›
824    by (simp add: RN "oth-class-taut:3:a" "universal-cor")
825  ultimately AOT_show ‹□∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn)›
826    using "→E" by blast
827qed
828
829AOT_theorem "id-rel-nec-equiv:2": ‹φ = ψ → □(φ ≡ ψ)›
830proof(rule "→I")
831  AOT_assume assumption: ‹φ = ψ›
832  AOT_hence ‹φ↓› and ‹ψ↓›
833    using "t=t-proper:1" "t=t-proper:2" MP by blast+
834  moreover AOT_have ‹∀p∀q (p = q → ((□(p ≡ p) → □(p ≡ q))))›
835    apply (rule GEN)+ using "l-identity"[axiom_inst] by force
836  ultimately AOT_have ‹φ = ψ → (□(φ ≡ φ) → □(φ ≡ ψ))›
837    using "∀E"(1) by blast
838  AOT_hence ‹□(φ ≡ φ) → □(φ ≡ ψ)›
839    using assumption "→E" by blast
840  moreover AOT_have ‹□(φ ≡ φ)›
841    by (simp add: RN "oth-class-taut:3:a" "universal-cor")
842  ultimately AOT_show ‹□(φ ≡ ψ)›
843    using "→E" by blast
844qed
845
846AOT_theorem "rule=E": assumes ‹φ{τ}› and ‹τ = σ› shows ‹φ{σ}›
847proof -
848  AOT_have ‹τ↓› and ‹σ↓› using assms(2) "t=t-proper:1" "t=t-proper:2" "→E" by blast+
849  moreover AOT_have ‹∀α∀β(α = β → (φ{α} → φ{β}))›
850    apply (rule GEN)+ using "l-identity"[axiom_inst] by blast
851  ultimately AOT_have ‹τ = σ → (φ{τ} → φ{σ})›
852    using "∀E"(1) by blast
853  AOT_thus ‹φ{σ}› using assms "→E" by blast
854qed
855
856AOT_theorem "propositions-lemma:1": ‹[λ φ] = φ›
857proof -
858  AOT_have ‹φ↓› by (simp add: "log-prop-prop:2")
859  moreover AOT_have ‹∀p [λ p] = p› using "lambda-predicates:3[zero]"[axiom_inst] "∀I" by fast
860  ultimately AOT_show ‹[λ φ] = φ›
861    using "∀E" by blast
862qed
863
864AOT_theorem "propositions-lemma:2": ‹[λ φ] ≡ φ›
865proof -
866  AOT_have ‹[λ φ] ≡ [λ φ]› by (simp add: "oth-class-taut:3:a")
867  AOT_thus ‹[λ φ] ≡ φ› using "propositions-lemma:1" "rule=E" by blast
868qed
869
870(* propositions-lemma:3 through propositions-lemma:5 do not apply *)
871
872AOT_theorem "propositions-lemma:6": ‹(φ ≡ ψ) ≡ ([λ φ] ≡ [λ ψ])›
873  by (metis "≡E"(1) "≡E"(5) "Associativity of ≡" "propositions-lemma:2")
874
875(* dr-alphabetic-rules does not apply *)
876
877AOT_theorem "oa-exist:1": ‹O!↓›
878proof -
879  AOT_have ‹[λx ◇[E!]x]↓› by "cqt:2[lambda]"
880  AOT_hence 1: ‹O! = [λx ◇[E!]x]› using "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1)] "→E" by blast
881  AOT_show ‹O!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
882qed
883
884AOT_theorem "oa-exist:2": ‹A!↓›
885proof -
886  AOT_have ‹[λx ¬◇[E!]x]↓› by "cqt:2[lambda]"
887  AOT_hence 1: ‹A! = [λx ¬◇[E!]x]› using "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1)] "→E" by blast
888  AOT_show ‹A!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
889qed
890
891AOT_theorem "oa-exist:3": ‹O!x ∨ A!x›
892proof(rule "raa-cor:1")
893  AOT_assume ‹¬(O!x ∨ A!x)›
894  AOT_hence A: ‹¬O!x› and B: ‹¬A!x›
895    using "Disjunction Addition"(1) "modus-tollens:1" "∨I"(2) "raa-cor:5" by blast+
896  AOT_have C: ‹O! = [λx ◇[E!]x]›
897    by (rule "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
898  AOT_have D: ‹A! = [λx ¬◇[E!]x]›
899    by (rule "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
900  AOT_have E: ‹¬[λx ◇[E!]x]x›
901    using A C "rule=E" by fast
902  AOT_have F: ‹¬[λx ¬◇[E!]x]x›
903    using B D "rule=E" by fast
904  AOT_have G: ‹[λx ◇[E!]x]x ≡ ◇[E!]x›
905    by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
906  AOT_have H: ‹[λx ¬◇[E!]x]x ≡ ¬◇[E!]x›
907    by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
908  AOT_show ‹¬◇[E!]x & ¬¬◇[E!]x› using G E "≡E" H F "≡E" "&I" by metis
909qed
910
911AOT_theorem "p-identity-thm2:1": ‹F = G ≡ □∀x(x[F] ≡ x[G])›
912proof -
913  AOT_have ‹F = G ≡ F↓ & G↓ & □∀x(x[F] ≡ x[G])›
914    using "identity:2" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
915  moreover AOT_have ‹F↓› and ‹G↓›
916    by (auto simp: "cqt:2[const_var]"[axiom_inst])
917  ultimately AOT_show ‹F = G ≡ □∀x(x[F] ≡ x[G])›
918    using "≡S"(1) "&I" by blast
919qed
920
921AOT_theorem "p-identity-thm2:2[2]": ‹F = G ≡ ∀y1([λx [F]xy1] = [λx [G]xy1] & [λx [F]y1x] = [λx [G]y1x])›
922proof -
923  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y1([λx [F]xy1] = [λx [G]xy1] & [λx [F]y1x] = [λx [G]y1x])›
924    using "identity:3[2]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
925  moreover AOT_have ‹F↓› and ‹G↓›
926    by (auto simp: "cqt:2[const_var]"[axiom_inst])
927  ultimately show ?thesis
928    using "≡S"(1) "&I" by blast
929qed
930    
931AOT_theorem "p-identity-thm2:2[3]": ‹F = G ≡ ∀y1∀y2([λx [F]xy1y2] = [λx [G]xy1y2] & [λx [F]y1xy2] = [λx [G]y1xy2] & [λx [F]y1y2x] = [λx [G]y1y2x])›
932proof -
933  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y1∀y2([λx [F]xy1y2] = [λx [G]xy1y2] & [λx [F]y1xy2] = [λx [G]y1xy2] & [λx [F]y1y2x] = [λx [G]y1y2x])›
934    using "identity:3[3]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
935  moreover AOT_have ‹F↓› and ‹G↓›
936    by (auto simp: "cqt:2[const_var]"[axiom_inst])
937  ultimately show ?thesis
938    using "≡S"(1) "&I" by blast
939qed
940
941AOT_theorem "p-identity-thm2:2[4]": ‹F = G ≡ ∀y1∀y2∀y3([λx [F]xy1y2y3] = [λx [G]xy1y2y3] & [λx [F]y1xy2y3] = [λx [G]y1xy2y3] & [λx [F]y1y2xy3] = [λx [G]y1y2xy3] & [λx [F]y1y2y3x] = [λx [G]y1y2y3x])›
942proof -
943  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y1∀y2∀y3([λx [F]xy1y2y3] = [λx [G]xy1y2y3] & [λx [F]y1xy2y3] = [λx [G]y1xy2y3] & [λx [F]y1y2xy3] = [λx [G]y1y2xy3] & [λx [F]y1y2y3x] = [λx [G]y1y2y3x])›
944    using "identity:3[4]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
945  moreover AOT_have ‹F↓› and ‹G↓›
946    by (auto simp: "cqt:2[const_var]"[axiom_inst])
947  ultimately show ?thesis
948    using "≡S"(1) "&I" by blast
949qed
950
951AOT_theorem "p-identity-thm2:2":
952  ‹F = G ≡ ∀x1...∀xn «AOT_sem_proj_id x1xn (λ τ . «[F]τ») (λ τ . «[G]τ»)»›
953proof -
954  AOT_have ‹F = G ≡ F↓ & G↓ & ∀x1...∀xn «AOT_sem_proj_id x1xn (λ τ . «[F]τ») (λ τ . «[G]τ»)»›
955    using "identity:3" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
956  moreover AOT_have ‹F↓› and ‹G↓›
957    by (auto simp: "cqt:2[const_var]"[axiom_inst])
958  ultimately show ?thesis
959    using "≡S"(1) "&I" by blast
960qed
961
962AOT_theorem "p-identity-thm2:3":
963  ‹p = q ≡ [λx p] = [λx q]›
964proof -
965  AOT_have ‹p = q ≡ p↓ & q↓ & [λx p] = [λx q]›
966    using "identity:4" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
967  moreover AOT_have ‹p↓› and ‹q↓›
968    by (auto simp: "cqt:2[const_var]"[axiom_inst])
969  ultimately show ?thesis
970    using "≡S"(1) "&I" by blast
971qed
972
973class AOT_Term_id_2 = AOT_Term_id + assumes "id-eq:1": ‹[v ⊨ α = α]›
974
975instance κ :: AOT_Term_id_2
976proof
977  AOT_modally_strict {
978    fix x
979    {
980      AOT_assume ‹O!x›
981      moreover AOT_have ‹□∀F([F]x ≡ [F]x)›
982        using RN GEN "oth-class-taut:3:a" by fast
983      ultimately AOT_have ‹O!x & O!x & □∀F([F]x ≡ [F]x)› using "&I" by simp
984    }
985    moreover {
986      AOT_assume ‹A!x›
987      moreover AOT_have ‹□∀F(x[F] ≡ x[F])›
988        using RN GEN "oth-class-taut:3:a" by fast
989      ultimately AOT_have ‹A!x & A!x & □∀F(x[F] ≡ x[F])› using "&I" by simp
990    }
991    ultimately AOT_have ‹(O!x & O!x & □∀F([F]x ≡ [F]x)) ∨ (A!x & A!x & □∀F(x[F] ≡ x[F]))›
992      using "oa-exist:3" "∨I"(1) "∨I"(2) "∨E"(3) "raa-cor:1" by blast
993    AOT_thus ‹x = x›
994      using "identity:1"[THEN "df-rules-formulas[4]"] "→E" by blast
995  }
996qed
997
998instance rel :: ("{AOT_κs,AOT_Term_id_2}") AOT_Term_id_2
999proof
1000  AOT_modally_strict {
1001    fix F :: "<'a> AOT_var"
1002    AOT_have 0: ‹[λx1...xn [F]x1...xn] = F›
1003      by (simp add: "lambda-predicates:3"[axiom_inst])
1004    AOT_have ‹[λx1...xn [F]x1...xn]↓›
1005      by "cqt:2[lambda]"
1006    AOT_hence ‹[λx1...xn [F]x1...xn] = [λx1...xn [F]x1...xn]›
1007      using "lambda-predicates:1"[axiom_inst] "→E" by blast
1008    AOT_show ‹F = F› using "rule=E" 0 by force 
1009  }
1010qed
1011
1012instance 𝗈 :: AOT_Term_id_2
1013proof
1014  AOT_modally_strict {
1015    fix p
1016    AOT_have 0: ‹[λ p] = p›
1017      by (simp add: "lambda-predicates:3[zero]"[axiom_inst])
1018    AOT_have ‹[λ p]↓›
1019      by (rule "cqt:2[lambda0]"[axiom_inst])
1020    AOT_hence ‹[λ p] = [λ p]›
1021      using "lambda-predicates:1[zero]"[axiom_inst] "→E" by blast
1022    AOT_show ‹p = p› using "rule=E" 0 by force
1023  }
1024qed
1025
1026instance prod :: (AOT_Term_id_2, AOT_Term_id_2) AOT_Term_id_2
1027proof
1028  AOT_modally_strict {
1029    fix α :: ‹('a×'b) AOT_var›
1030    AOT_show ‹α = α›
1031    proof (induct)
1032      AOT_show ‹τ = τ› if ‹τ↓› for τ :: ‹'a×'b›
1033        using that
1034      proof (induct τ)
1035        fix τ1 :: 'a and τ2 :: 'b
1036        AOT_assume ‹«(τ1,τ2)»↓›
1037        AOT_hence ‹τ1↓› and ‹τ2↓› using "≡dfE" "&E" tuple_denotes by blast+
1038        AOT_hence ‹τ1 = τ1› and ‹τ2 = τ2› using "id-eq:1"[unvarify α] by blast+
1039        AOT_thus ‹«(τ1, τ2)» = «(τ1, τ2)»› by (metis "≡dfI" "&I" tuple_identity_1)
1040      qed
1041    qed
1042  }
1043qed
1044
1045AOT_register_type_constraints
1046  Term: ‹_::AOT_Term_id_2› ‹_::AOT_Term_id_2›
1047AOT_register_type_constraints
1048  Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id_2}›
1049AOT_register_type_constraints
1050  Relation: ‹<_::{AOT_κs, AOT_Term_id_2}>›
1051
1052(* TODO: Interestingly, this doesn't depend on "id-eq:1" at all! *)
1053AOT_theorem "id-eq:2": ‹α = β → β = α›
1054(*
1055  TODO: look at this proof generated using:
1056        including AOT_no_atp sledgehammer[isar_proofs = true]
1057proof -
1058  have "(∃φ. [v ⊨ ~β = α → ~φ] ∧ [v ⊨ α = β → φ]) ∨ (∃φ. ¬ [v ⊨ φ{α} → φ{β}])"
1059    by meson
1060  then show ?thesis
1061    by (meson "contraposition:2" "Hypothetical Syllogism" "deduction-theorem" l_"identity:1" "useful-tautologies:1")
1062qed
1063*)
1064(*  by (meson "rule=E" "deduction-theorem") *)
1065proof (rule "→I")
1066  AOT_assume ‹α = β›
1067  moreover AOT_have ‹β = β› using calculation "rule=E"[of _ "λ τ . «τ = β»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1068  moreover AOT_have ‹α = α → α = α› using "if-p-then-p" by blast
1069  ultimately AOT_show ‹β = α›
1070    using "→E" "→I" "rule=E"[of _ "λ τ . «(τ = τ) → (τ = α)»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1071qed
1072
1073AOT_theorem "id-eq:3": ‹α = β & β = γ → α = γ›
1074  using "rule=E" "→I" "&E" by blast
1075
1076AOT_theorem "id-eq:4": ‹α = β ≡ ∀γ (α = γ ≡ β = γ)›
1077proof (rule "≡I"; rule "→I")
1078  AOT_assume 0: ‹α = β›
1079  AOT_hence 1: ‹β = α› using "id-eq:2" "→E" by blast
1080  AOT_show ‹∀γ (α = γ ≡ β = γ)›
1081    by (rule GEN) (metis "≡I" "→I" 0 "1" "rule=E")
1082next
1083  AOT_assume ‹∀γ (α = γ ≡ β = γ)›
1084  AOT_hence ‹α = α ≡ β = α› using "∀E"(2) by blast
1085  AOT_hence ‹α = α → β = α› using "≡E"(1) "→I" by blast
1086  AOT_hence ‹β = α› using "id-eq:1" "→E" by blast
1087  AOT_thus ‹α = β› using "id-eq:2" "→E" by blast
1088qed
1089
1090AOT_theorem "rule=I:1": assumes ‹τ↓› shows ‹τ = τ›
1091proof -
1092  AOT_have ‹∀α (α = α)›
1093    by (rule GEN) (metis "id-eq:1")
1094  AOT_thus ‹τ = τ› using assms "∀E" by blast
1095qed
1096
1097AOT_theorem "rule=I:2[const_var]": "α = α"
1098  using "id-eq:1".
1099
1100AOT_theorem "rule=I:2[lambda]": assumes ‹INSTANCE_OF_CQT_2(φ)› shows "[λν1...νn φ{ν1...νn}] = [λν1...νn φ{ν1...νn}]"
1101proof -
1102  AOT_have ‹∀α (α = α)›
1103    by (rule GEN) (metis "id-eq:1")
1104  moreover AOT_have ‹[λν1...νn φ{ν1...νn}]↓› using assms by (rule "cqt:2[lambda]"[axiom_inst])
1105  ultimately AOT_show ‹[λν1...νn φ{ν1...νn}] = [λν1...νn φ{ν1...νn}]› using assms "∀E" by blast
1106qed
1107
1108lemmas "=I" = "rule=I:1" "rule=I:2[const_var]" "rule=I:2[lambda]"
1109
1110AOT_theorem "rule-id-df:1":
1111  assumes ‹τ{α1...αn} =df σ{α1...αn}› and ‹σ{τ1...τn}↓›
1112  shows ‹τ{τ1...τn} = σ{τ1...τn}›
1113proof -
1114  AOT_have ‹σ{τ1...τn}↓ → τ{τ1...τn} = σ{τ1...τn}›
1115    using "df-rules-terms[3]" assms(1) "&E" by blast
1116  AOT_thus ‹τ{τ1...τn} = σ{τ1...τn}›
1117    using assms(2) "→E" by blast
1118qed
1119
1120AOT_theorem "rule-id-df:1[zero]":
1121  assumes ‹τ =df σ› and ‹σ↓›
1122  shows ‹τ = σ›
1123proof -
1124  AOT_have ‹σ↓ → τ = σ›
1125    using "df-rules-terms[4]" assms(1) "&E" by blast
1126  AOT_thus ‹τ = σ›
1127    using assms(2) "→E" by blast
1128qed
1129
1130AOT_theorem "rule-id-df:2:a":
1131  assumes ‹τ{α1...αn} =df σ{α1...αn}› and ‹σ{τ1...τn}↓› and ‹φ{τ{τ1...τn}}›
1132  shows ‹φ{σ{τ1...τn}}›
1133proof -
1134  AOT_have ‹τ{τ1...τn} = σ{τ1...τn}› using "rule-id-df:1" assms(1,2) by blast
1135  AOT_thus ‹φ{σ{τ1...τn}}› using assms(3) "rule=E" by blast
1136qed
1137
1138(* TODO: get rid of this, ideally *)
1139AOT_theorem "rule-id-df:2:a[2]":
1140  assumes ‹τ{«(α1,α2)»} =df σ{«(α1,α2)»}› and ‹σ{«(τ1,τ2)»}↓› and ‹φ{τ{«(τ1,τ2)»}}›
1141  shows ‹φ{σ{«(τ1,τ2)»}}›
1142proof -
1143  AOT_have ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1144  proof -
1145    AOT_have ‹σ{«(τ1,τ2)»}↓ → τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1146      using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
1147    AOT_thus ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1148      using assms(2) "→E" by blast
1149  qed
1150  AOT_thus ‹φ{σ{«(τ1,τ2)»}}› using assms(3) "rule=E" by blast
1151qed
1152
1153AOT_theorem "rule-id-df:2:a[zero]":
1154  assumes ‹τ =df σ› and ‹σ↓› and ‹φ{τ}›
1155  shows ‹φ{σ}›
1156proof -
1157  AOT_have ‹τ = σ› using "rule-id-df:1[zero]" assms(1,2) by blast
1158  AOT_thus ‹φ{σ}› using assms(3) "rule=E" by blast
1159qed
1160
1161lemmas "=dfE" = "rule-id-df:2:a" "rule-id-df:2:a[zero]"
1162
1163AOT_theorem "rule-id-df:2:b":
1164  assumes ‹τ{α1...αn} =df σ{α1...αn}› and ‹σ{τ1...τn}↓› and ‹φ{σ{τ1...τn}}›
1165  shows ‹φ{τ{τ1...τn}}›
1166proof -
1167  AOT_have ‹τ{τ1...τn} = σ{τ1...τn}› using "rule-id-df:1" assms(1,2) by blast
1168  AOT_hence ‹σ{τ1...τn} = τ{τ1...τn}›
1169    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1170  AOT_thus ‹φ{τ{τ1...τn}}› using assms(3) "rule=E" by blast
1171qed
1172
1173(* TODO: get rid of this, ideally *)
1174AOT_theorem "rule-id-df:2:b[2]":
1175  assumes ‹τ{«(α1,α2)»} =df σ{«(α1,α2)»}› and ‹σ{«(τ1,τ2)»}↓› and ‹φ{σ{«(τ1,τ2)»}}›
1176  shows ‹φ{τ{«(τ1,τ2)»}}›
1177proof -
1178  AOT_have ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1179  proof -
1180    AOT_have ‹σ{«(τ1,τ2)»}↓ → τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1181      using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
1182    AOT_thus ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1183      using assms(2) "→E" by blast
1184  qed
1185  AOT_hence ‹σ{«(τ1,τ2)»} = τ{«(τ1,τ2)»}›
1186    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1187  AOT_thus ‹φ{τ{«(τ1,τ2)»}}› using assms(3) "rule=E" by blast
1188qed
1189
1190AOT_theorem "rule-id-df:2:b[zero]":
1191  assumes ‹τ =df σ› and ‹σ↓› and ‹φ{σ}›
1192  shows ‹φ{τ}›
1193proof -
1194  AOT_have ‹τ = σ› using "rule-id-df:1[zero]" assms(1,2) by blast
1195  AOT_hence ‹σ = τ›
1196    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1197  AOT_thus ‹φ{τ}› using assms(3) "rule=E" by blast
1198qed
1199
1200lemmas "=dfI" = "rule-id-df:2:b" "rule-id-df:2:b[zero]"
1201
1202AOT_theorem "free-thms:1": ‹τ↓ ≡ ∃β (β = τ)›
1203  by (metis "∃E" "rule=I:1" "t=t-proper:2" "→I" "∃I"(1) "≡I" "→E")
1204
1205AOT_theorem "free-thms:2": ‹∀α φ{α} → (∃β (β = τ) → φ{τ})›
1206  by (metis "∃E" "rule=E" "cqt:2[const_var]"[axiom_inst] "→I" "∀E"(1))
1207
1208AOT_theorem "free-thms:3[const_var]": ‹∃β (β = α)›
1209  by (meson "∃I"(2) "id-eq:1")
1210
1211AOT_theorem "free-thms:3[lambda]": assumes ‹INSTANCE_OF_CQT_2(φ)› shows ‹∃β (β = [λν1...νn φ{ν1...νn}])›
1212  by (meson "=I"(3) assms "cqt:2[lambda]"[axiom_inst] "existential:1")
1213
1214AOT_theorem "free-thms:4[rel]": ‹([Π]κ1...κn ∨ κ1...κn[Π]) → ∃β (β = Π)›
1215  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1216
1217(* TODO: this is a rather weird way to formulate this and we don't have tuple-existential-elimination
1218         or tuple-equality-elimination in the theory... Splitting them is also a bit unfortunate, though.*)
1219AOT_theorem "free-thms:4[vars]": ‹([Π]κ1...κn ∨ κ1...κn[Π]) → ∃β1...∃βn (β1...βn = κ1...κn)›
1220  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1221
1222AOT_theorem "free-thms:4[1,rel]": ‹([Π]κ ∨ κ[Π]) → ∃β (β = Π)›
1223  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1224AOT_theorem "free-thms:4[1,1]": ‹([Π]κ ∨ κ[Π]) → ∃β (β = κ)›
1225  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1226
1227AOT_theorem "free-thms:4[2,rel]": ‹([Π]κ1κ2 ∨ κ1κ2[Π]) → ∃β (β = Π)›
1228  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1229AOT_theorem "free-thms:4[2,1]": ‹([Π]κ1κ2 ∨ κ1κ2[Π]) → ∃β (β = κ1)›
1230  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1231AOT_theorem "free-thms:4[2,2]": ‹([Π]κ1κ2 ∨ κ1κ2[Π]) → ∃β (β = κ2)›
1232  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1233AOT_theorem "free-thms:4[3,rel]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = Π)›
1234  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1235AOT_theorem "free-thms:4[3,1]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = κ1)›
1236  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1237AOT_theorem "free-thms:4[3,2]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = κ2)›
1238  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1239AOT_theorem "free-thms:4[3,3]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = κ3)›
1240  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1241AOT_theorem "free-thms:4[4,rel]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = Π)›
1242  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1243AOT_theorem "free-thms:4[4,1]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ1)›
1244  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1245AOT_theorem "free-thms:4[4,2]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ2)›
1246  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1247AOT_theorem "free-thms:4[4,3]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ3)›
1248  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1249AOT_theorem "free-thms:4[4,4]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ4)›
1250  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1251
1252AOT_theorem "ex:1:a": ‹∀α α↓›
1253  by (rule GEN) (fact "cqt:2[const_var]"[axiom_inst])
1254AOT_theorem "ex:1:b": ‹∀α∃β(β = α)›
1255  by (rule GEN) (fact "free-thms:3[const_var]")
1256
1257AOT_theorem "ex:2:a": ‹□α↓›
1258  by (rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1259AOT_theorem "ex:2:b": ‹□∃β(β = α)›
1260  by (rule RN) (fact "free-thms:3[const_var]")
1261
1262AOT_theorem "ex:3:a": ‹□∀α α↓›
1263  by (rule RN) (fact "ex:1:a")
1264AOT_theorem "ex:3:b": ‹□∀α∃β(β = α)›
1265  by (rule RN) (fact "ex:1:b")
1266
1267AOT_theorem "ex:4:a": ‹∀α □α↓›
1268  by (rule GEN; rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1269AOT_theorem "ex:4:b": ‹∀α□∃β(β = α)›
1270  by (rule GEN; rule RN) (fact "free-thms:3[const_var]")
1271
1272AOT_theorem "ex:5:a": ‹□∀α □α↓›
1273  by (rule RN) (simp add: "ex:4:a")
1274AOT_theorem "ex:5:b": ‹□∀α□∃β(β = α)›
1275  by (rule RN) (simp add: "ex:4:b")
1276
1277AOT_theorem "all-self=:1": ‹□∀α(α = α)›
1278  by (rule RN; rule GEN) (fact "id-eq:1")
1279AOT_theorem "all-self=:2": ‹∀α□(α = α)›
1280  by (rule GEN; rule RN) (fact "id-eq:1")
1281
1282AOT_theorem "id-nec:1": ‹α = β → □(α = β)›
1283proof(rule "→I")
1284  AOT_assume ‹α = β›
1285  moreover AOT_have ‹□(α = α)›
1286    by (rule RN) (fact "id-eq:1")
1287  ultimately AOT_show ‹□(α = β)› using "rule=E" by fast
1288qed
1289
1290AOT_theorem "id-nec:2": ‹τ = σ → □(τ = σ)›
1291proof(rule "→I")
1292  AOT_assume asm: ‹τ = σ›
1293  moreover AOT_have ‹τ↓›
1294    using calculation "t=t-proper:1" "→E" by blast
1295  moreover AOT_have ‹□(τ = τ)›
1296    using calculation "all-self=:2" "∀E"(1) by blast
1297  ultimately AOT_show ‹□(τ = σ)› using "rule=E" by fast
1298qed
1299
1300AOT_theorem "term-out:1": ‹φ{α} ≡ ∃β (β = α & φ{β})›
1301proof (rule "≡I"; rule "→I")
1302  AOT_assume asm: ‹φ{α}›
1303  AOT_show ‹∃β (β = α & φ{β})›
1304    by (rule "∃I"(2)[where β=α]; rule "&I")
1305       (auto simp: "id-eq:1" asm)
1306next
1307  AOT_assume 0: ‹∃β (β = α & φ{β})›
1308  (* TODO: have another look at this instantiation. Ideally AOT_obtain would resolve directly to the existential
1309           statement as proof obligation *)
1310  AOT_obtain β where ‹β = α & φ{β}› using "instantiation"[rotated, OF 0] by blast
1311  AOT_thus ‹φ{α}› using "&E" "rule=E" by blast
1312qed
1313
1314AOT_theorem "term-out:2": ‹τ↓ → (φ{τ} ≡ ∃α(α = τ & φ{α}))›
1315proof(rule "→I")
1316  AOT_assume ‹τ↓›
1317  moreover AOT_have ‹∀α (φ{α} ≡ ∃β (β = α & φ{β}))›
1318    by (rule GEN) (fact "term-out:1")
1319  ultimately AOT_show ‹φ{τ} ≡ ∃α(α = τ & φ{α})›
1320    using "∀E" by blast
1321qed
1322
1323(* TODO: example of an apply-style proof. Keep or reformulate? *)
1324AOT_theorem "term-out:3": ‹(φ{α} & ∀β(φ{β} → β = α)) ≡ ∀β(φ{β} ≡ β = α)›
1325  apply (rule "≡I"; rule "→I")
1326   apply (frule "&E"(1)) apply (drule "&E"(2))
1327   apply (rule GEN; rule "≡I"; rule "→I")
1328  using "rule-ui:2[const_var]" "vdash-properties:5" apply blast
1329  apply (meson "rule=E" "id-eq:1")
1330  apply (rule "&I")
1331  using "id-eq:1" "≡E"(2) "rule-ui:3" apply blast
1332  apply (rule GEN; rule "→I")
1333  using "≡E"(1) "rule-ui:2[const_var]" by blast
1334
1335AOT_theorem "term-out:4": ‹(φ{β} & ∀α(φ{α} → α = β)) ≡ ∀α(φ{α} ≡ α = β)›
1336  using "term-out:3" . (* TODO: same as above - another instance of the generalized alphabetic variant rule... *)
1337
1338(* TODO: would of course be nice to define it without the syntax magic *)
1339AOT_define AOT_exists_unique :: ‹α ⇒ φ ⇒ φ›
1340  "uniqueness:1": ‹«AOT_exists_unique φ» ≡df ∃α (φ{α} & ∀β (φ{β} → β = α))›
1341syntax "_AOT_exists_unique" :: ‹α ⇒ φ ⇒ φ› ("∃!_ _" [1,40])
1342AOT_syntax_print_translations
1343  "_AOT_exists_unique τ φ" <= "CONST AOT_exists_unique (_abs τ φ)"
1344syntax
1345   "_AOT_exists_unique_ellipse" :: ‹id_position ⇒ id_position ⇒ φ ⇒ φ› (‹∃!_...∃!_ _› [1,40])
1346parse_ast_translation‹[(syntax_const‹_AOT_exists_unique_ellipse›, fn ctx => fn [a,b,c] =>
1347  Ast.mk_appl (Ast.Constant "AOT_exists_unique") [parseEllipseList "_AOT_vars" ctx [a,b],c]),
1348(syntax_const‹_AOT_exists_unique›,AOT_restricted_binder const_name‹AOT_exists_unique› const_syntax‹AOT_conj›)]›
1349print_translation‹AOT_syntax_print_translations
1350  [AOT_preserve_binder_abs_tr' const_syntax‹AOT_exists_unique› syntax_const‹_AOT_exists_unique› (syntax_const‹_AOT_exists_unique_ellipse›, true) const_name‹AOT_conj›,
1351  AOT_binder_trans @{theory} @{binding "AOT_exists_unique_binder"} syntax_const‹_AOT_exists_unique›]
1352›
1353
1354
1355context AOT_meta_syntax
1356begin
1357notation AOT_exists_unique (binder "∃!" 20)
1358end
1359context AOT_no_meta_syntax
1360begin
1361no_notation AOT_exists_unique (binder "∃!" 20)
1362end
1363
1364AOT_theorem "uniqueness:2": ‹∃!α φ{α} ≡ ∃α∀β(φ{β} ≡ β = α)›
1365proof(rule "≡I"; rule "→I")
1366    AOT_assume ‹∃!α φ{α}›
1367    AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
1368      using "uniqueness:1" "≡dfE" by blast
1369    then AOT_obtain α where ‹φ{α} & ∀β (φ{β} → β = α)› using "instantiation"[rotated] by blast
1370    AOT_hence ‹∀β(φ{β} ≡ β = α)› using "term-out:3" "≡E" by blast
1371    AOT_thus ‹∃α∀β(φ{β} ≡ β = α)›
1372      using "∃I" by fast
1373next
1374    AOT_assume ‹∃α∀β(φ{β} ≡ β = α)›
1375    then AOT_obtain α where ‹∀β (φ{β} ≡ β = α)› using "instantiation"[rotated] by blast
1376    AOT_hence ‹φ{α} & ∀β (φ{β} → β = α)› using "term-out:3" "≡E" by blast
1377    AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
1378      using "∃I" by fast
1379    AOT_thus ‹∃!α φ{α}›
1380      using "uniqueness:1" "≡dfI" by blast
1381qed
1382
1383AOT_theorem "uni-most": ‹∃!α φ{α} → ∀β∀γ((φ{β} & φ{γ}) → β = γ)›
1384proof(rule "→I"; rule GEN; rule GEN; rule "→I")
1385  fix β γ
1386  AOT_assume ‹∃!α φ{α}›
1387  AOT_hence ‹∃α∀β(φ{β} ≡ β = α)›
1388    using "uniqueness:2" "≡E" by blast
1389  then AOT_obtain α where ‹∀β(φ{β} ≡ β = α)›
1390    using "instantiation"[rotated] by blast
1391  moreover AOT_assume ‹φ{β} & φ{γ}›
1392  ultimately AOT_have ‹β = α› and ‹γ = α›
1393    using "∀E"(2) "&E" "≡E"(1,2) by blast+
1394  AOT_thus ‹β = γ›
1395    by (metis "rule=E" "id-eq:2" "→E")
1396qed
1397
1398AOT_theorem "nec-exist-!": ‹∀α(φ{α} → □φ{α}) → (∃!α φ{α} → ∃!α □φ{α})›
1399proof (rule "→I"; rule "→I")
1400  AOT_assume a: ‹∀α(φ{α} → □φ{α})›
1401  AOT_assume ‹∃!α φ{α}›
1402  AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))› using "uniqueness:1" "≡dfE" by blast
1403  then AOT_obtain α where ξ: ‹φ{α} & ∀β (φ{β} → β = α)› using "instantiation"[rotated] by blast
1404  AOT_have ‹□φ{α}›
1405    using ξ a "&E" "∀E" "→E" by fast
1406  moreover AOT_have ‹∀β (□φ{β} → β = α)›
1407    apply (rule GEN; rule "→I")
1408    using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
1409  ultimately AOT_have ‹(□φ{α} & ∀β (□φ{β} → β = α))›
1410    using "&I" by blast
1411  AOT_thus ‹∃!α □φ{α}›
1412    using "uniqueness:1" "≡dfI" "∃I" by fast
1413qed
1414
1415AOT_theorem "act-cond": ‹𝒜(φ → ψ) → (𝒜φ → 𝒜ψ)›
1416  using "→I" "≡E"(1) "logic-actual-nec:2"[axiom_inst] by blast
1417
1418AOT_theorem "nec-imp-act": ‹□φ → 𝒜φ›
1419  by (metis "act-cond" "contraposition:1[2]" "≡E"(4) "qml:2"[THEN act_closure, axiom_inst] "qml-act:2"[axiom_inst] RAA(1) "→E" "→I")
1420
1421AOT_theorem "act-conj-act:1": ‹𝒜(𝒜φ → φ)›
1422  using "→I" "≡E"(2) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] by blast
1423
1424AOT_theorem "act-conj-act:2": ‹𝒜(φ → 𝒜φ)›
1425  by (metis "→I" "≡E"(2, 4) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] RAA(1))
1426
1427AOT_theorem "act-conj-act:3": ‹(𝒜φ & 𝒜ψ) → 𝒜(φ & ψ)›
1428proof -
1429  AOT_have ‹□(φ → (ψ → (φ & ψ)))›
1430    by (rule RN) (fact Adjunction)
1431  AOT_hence ‹𝒜(φ → (ψ → (φ & ψ)))›
1432    using "nec-imp-act" "→E" by blast
1433  AOT_hence ‹𝒜φ → 𝒜(ψ → (φ & ψ))›
1434    using "act-cond" "→E" by blast
1435  moreover AOT_have ‹𝒜(ψ → (φ & ψ)) → (𝒜ψ → 𝒜(φ & ψ))›
1436    by (fact "act-cond")
1437  ultimately AOT_have ‹𝒜φ → (𝒜ψ → 𝒜(φ & ψ))›
1438    using "→I" "→E" by metis
1439  AOT_thus ‹(𝒜φ & 𝒜ψ) → 𝒜(φ & ψ)›
1440    by (metis Importation "→E")
1441qed
1442
1443AOT_theorem "act-conj-act:4": ‹𝒜(𝒜φ ≡ φ)›
1444proof -
1445  AOT_have ‹(𝒜(𝒜φ → φ) & 𝒜(φ → 𝒜φ)) → 𝒜((𝒜φ → φ) & (φ → 𝒜φ))›
1446    by (fact "act-conj-act:3")
1447  moreover AOT_have ‹𝒜(𝒜φ → φ) & 𝒜(φ → 𝒜φ)›
1448    using "&I" "act-conj-act:1" "act-conj-act:2" by simp
1449  ultimately AOT_have ζ: ‹𝒜((𝒜φ → φ) & (φ → 𝒜φ))›
1450    using "→E" by blast
1451  AOT_have ‹𝒜(((𝒜φ → φ) & (φ → 𝒜φ)) → (𝒜φ ≡ φ))›
1452    using "conventions:3"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst] by blast
1453  AOT_hence ‹𝒜((𝒜φ → φ) & (φ → 𝒜φ)) → 𝒜(𝒜φ ≡ φ)›
1454    using "act-cond" "→E" by blast
1455  AOT_thus ‹𝒜(𝒜φ ≡ φ)› using ζ "→E" by blast
1456qed
1457
1458(* TODO: consider introducing AOT_inductive *)
1459inductive arbitrary_actualization for φ where
1460  ‹arbitrary_actualization φ «𝒜φ»›
1461| ‹arbitrary_actualization φ «𝒜ψ»› if ‹arbitrary_actualization φ ψ›
1462declare arbitrary_actualization.cases[AOT] arbitrary_actualization.induct[AOT]
1463        arbitrary_actualization.simps[AOT] arbitrary_actualization.intros[AOT]
1464syntax arbitrary_actualization :: ‹φ' ⇒ φ' ⇒ AOT_prop› ("ARBITRARY'_ACTUALIZATION'(_,_')")
1465
1466notepad
1467begin
1468  AOT_modally_strict {
1469    fix φ
1470    AOT_have ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, 𝒜(𝒜φ ≡ φ))›
1471      using AOT_PLM.arbitrary_actualization.intros by metis
1472    AOT_have ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, 𝒜𝒜(𝒜φ ≡ φ))›
1473      using AOT_PLM.arbitrary_actualization.intros by metis
1474    AOT_have ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, 𝒜𝒜𝒜(𝒜φ ≡ φ))›
1475      using AOT_PLM.arbitrary_actualization.intros by metis
1476  }
1477end
1478
1479
1480AOT_theorem "closure-act:1": assumes ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, ψ)› shows ‹ψ›
1481using assms proof(induct)
1482  case 1
1483  AOT_show ‹𝒜(𝒜φ ≡ φ)›
1484    by (simp add: "act-conj-act:4")
1485next
1486  case (2 ψ)
1487  AOT_thus ‹𝒜ψ›
1488    by (metis arbitrary_actualization.simps "≡E"(1) "logic-actual-nec:4"[axiom_inst])
1489qed
1490
1491AOT_theorem "closure-act:2": ‹∀α 𝒜(𝒜φ{α} ≡ φ{α})›
1492  by (simp add: "act-conj-act:4" "∀I")
1493
1494AOT_theorem "closure-act:3": ‹𝒜∀α 𝒜(𝒜φ{α} ≡ φ{α})›
1495  by (metis (no_types, lifting) "act-conj-act:4" "≡E"(1,2) "logic-actual-nec:3"[axiom_inst] "logic-actual-nec:4"[axiom_inst] "∀I")
1496
1497AOT_theorem "closure-act:4": ‹𝒜∀α1...∀αn 𝒜(𝒜φ{α1...αn} ≡ φ{α1...αn})›
1498  using "closure-act:3" .
1499
1500(* TODO: examine these proofs *)
1501AOT_theorem "RA[1]": assumes ‹⊢ φ› shows ‹⊢ 𝒜φ›
1502  (* This proof is the one rejected in remark (136) (meta-rule) *)
1503  using "¬¬E" assms "≡E"(3) "logic-actual"[act_axiom_inst] "logic-actual-nec:1"[axiom_inst] "modus-tollens:2" by blast
1504AOT_theorem "RA[2]": assumes ‹⊢□ φ› shows ‹𝒜φ›
1505  (* This is actually Γ ⊢□ φ ⟹ □Γ ⊢□ 𝒜φ*)
1506  using RN assms "nec-imp-act" "vdash-properties:5" by blast
1507AOT_theorem "RA[3]": assumes ‹Γ ⊢□ φ› shows ‹𝒜Γ ⊢□ 𝒜φ›
1508  using assms by (meson AOT_sem_act imageI)
1509  (* This is not exactly right either. *)
1510
1511AOT_act_theorem "ANeg:1": ‹¬𝒜φ ≡ ¬φ›
1512  by (simp add: "RA[1]" "contraposition:1[1]" "deduction-theorem" "≡I" "logic-actual"[act_axiom_inst])
1513
1514AOT_act_theorem "ANeg:2": ‹¬𝒜¬φ ≡ φ›
1515  using "ANeg:1" "≡I" "≡E"(5) "useful-tautologies:1" "useful-tautologies:2" by blast
1516
1517AOT_theorem "Act-Basic:1": ‹𝒜φ ∨ 𝒜¬φ›
1518  by (meson "∨I"(1,2) "≡E"(2) "logic-actual-nec:1"[axiom_inst] "raa-cor:1")
1519
1520AOT_theorem "Act-Basic:2": ‹𝒜(φ & ψ) ≡ (𝒜φ & 𝒜ψ)›
1521proof (rule "≡I"; rule "→I")
1522  AOT_assume ‹𝒜(φ & ψ)›
1523  moreover AOT_have ‹𝒜((φ & ψ) → φ)›
1524    by (simp add: "RA[2]" "Conjunction Simplification"(1))
1525  moreover AOT_have ‹𝒜((φ & ψ) → ψ)›
1526    by (simp add: "RA[2]" "Conjunction Simplification"(2))
1527  ultimately AOT_show ‹𝒜φ & 𝒜ψ›
1528    using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1529next
1530  AOT_assume ‹𝒜φ & 𝒜ψ›
1531  AOT_thus ‹𝒜(φ & ψ)›
1532    using "act-conj-act:3" "vdash-properties:6" by blast
1533qed
1534
1535AOT_theorem "Act-Basic:3": ‹𝒜(φ ≡ ψ) ≡ (𝒜(φ → ψ) & 𝒜(ψ → φ))›
1536proof (rule "≡I"; rule "→I")
1537  AOT_assume ‹𝒜(φ ≡ ψ)›
1538  moreover AOT_have ‹𝒜((φ ≡ ψ) → (φ → ψ))›
1539    by (simp add: "RA[2]" "deduction-theorem" "≡E"(1))
1540  moreover AOT_have ‹𝒜((φ ≡ ψ) → (ψ → φ))›
1541    by (simp add: "RA[2]" "deduction-theorem" "≡E"(2))
1542  ultimately AOT_show ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1543    using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1544next
1545  AOT_assume ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1546  AOT_hence ‹𝒜((φ → ψ) & (ψ → φ))›
1547    by (metis "act-conj-act:3" "vdash-properties:10")
1548  moreover AOT_have ‹𝒜(((φ → ψ) & (ψ → φ)) → (φ ≡ ψ))›
1549    by (simp add: "conventions:3" "RA[2]" "df-rules-formulas[2]" "vdash-properties:1[2]")
1550  ultimately AOT_show ‹𝒜(φ ≡ ψ)›
1551    using "act-cond"[THEN "→E", THEN "→E"] by metis
1552qed
1553
1554AOT_theorem "Act-Basic:4": ‹(𝒜(φ → ψ) & 𝒜(ψ → φ)) ≡ (𝒜φ ≡ 𝒜ψ)›
1555proof (rule "≡I"; rule "→I")
1556  AOT_assume 0: ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1557  AOT_show ‹𝒜φ ≡ 𝒜ψ›
1558    using 0 "&E" "act-cond"[THEN "→E", THEN "→E"] "≡I" "→I" by metis
1559next
1560  AOT_assume ‹𝒜φ ≡ 𝒜ψ›
1561  AOT_thus ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1562    by (metis "→I" "logic-actual-nec:2"[axiom_inst] "≡E"(1,2) "&I")
1563qed
1564
1565AOT_theorem "Act-Basic:5": ‹𝒜(φ ≡ ψ) ≡ (𝒜φ ≡ 𝒜ψ)›
1566  using "Act-Basic:3" "Act-Basic:4" "≡E"(5) by blast
1567
1568AOT_theorem "Act-Basic:6": ‹𝒜φ ≡ □𝒜φ›
1569  by (simp add: "≡I" "qml:2"[axiom_inst] "qml-act:1"[axiom_inst])
1570
1571AOT_theorem "Act-Basic:7": ‹𝒜□φ → □𝒜φ›
1572  by (metis "Act-Basic:6" "→I" "→E" "≡E"(1,2) "nec-imp-act" "qml-act:2"[axiom_inst])
1573
1574AOT_theorem "Act-Basic:8": ‹□φ → □𝒜φ›
1575  using "Hypothetical Syllogism" "nec-imp-act" "qml-act:1"[axiom_inst] by blast
1576
1577AOT_theorem "Act-Basic:9": ‹𝒜(φ ∨ ψ) ≡ (𝒜φ ∨ 𝒜ψ)›
1578proof (rule "≡I"; rule "→I")
1579  AOT_assume ‹𝒜(φ ∨ ψ)›
1580  AOT_thus ‹𝒜φ ∨ 𝒜ψ›
1581  proof (rule "raa-cor:3")
1582    AOT_assume ‹¬(𝒜φ ∨ 𝒜ψ)›
1583    AOT_hence ‹¬𝒜φ & ¬𝒜ψ›
1584      by (metis "≡E"(1) "oth-class-taut:5:d")
1585    AOT_hence ‹𝒜¬φ & 𝒜¬ψ›
1586      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] "&E" "&I" by metis
1587    AOT_hence ‹𝒜(¬φ & ¬ψ)›
1588      using "≡E" "Act-Basic:2" by metis
1589    moreover AOT_have ‹𝒜((¬φ & ¬ψ) ≡ ¬(φ ∨ ψ))›
1590      using "RA[2]" "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:d" by blast
1591    moreover AOT_have ‹𝒜(¬φ & ¬ψ) ≡ 𝒜(¬(φ ∨ ψ))›
1592      using calculation(2) by (metis "Act-Basic:5" "≡E"(1))
1593    ultimately AOT_have ‹𝒜(¬(φ ∨ ψ))› using "≡E" by blast
1594    AOT_thus ‹¬𝒜(φ ∨ ψ)›
1595      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by auto
1596  qed
1597next
1598  AOT_assume ‹𝒜φ ∨ 𝒜ψ›
1599  AOT_thus ‹𝒜(φ ∨ ψ)›
1600    by (meson "RA[2]" "act-cond" "∨I"(1) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
1601qed
1602
1603AOT_theorem "Act-Basic:10": ‹𝒜∃α φ{α} ≡ ∃α 𝒜φ{α}›
1604proof -
1605  AOT_have θ: ‹¬𝒜∀α ¬φ{α} ≡ ¬∀α 𝒜¬φ{α}›
1606    by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1607       (metis "logic-actual-nec:3"[axiom_inst])
1608  AOT_have ξ: ‹¬∀α 𝒜¬φ{α} ≡ ¬∀α ¬𝒜φ{α}›
1609    by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1610       (rule "logic-actual-nec:1"[THEN universal_closure, axiom_inst, THEN "cqt-basic:3"[THEN "→E"]])
1611  AOT_have ‹𝒜(∃α φ{α}) ≡ 𝒜(¬∀α ¬φ{α})›
1612    using "conventions:4"[THEN "df-rules-formulas[1]", THEN act_closure, axiom_inst]
1613          "conventions:4"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst]
1614    "Act-Basic:4"[THEN "≡E"(1)] "&I" "Act-Basic:5"[THEN "≡E"(2)] by metis
1615  also AOT_have ‹… ≡ ¬𝒜∀α ¬φ{α}›
1616    by (simp add: "logic-actual-nec:1" "vdash-properties:1[2]")
1617  also AOT_have ‹… ≡ ¬∀α 𝒜 ¬φ{α}› using θ by blast
1618  also AOT_have ‹… ≡ ¬∀α ¬𝒜 φ{α}› using ξ by blast
1619  also AOT_have ‹… ≡ ∃α 𝒜 φ{α}›
1620    using "conventions:4"[THEN "≡Df"] by (metis "≡E"(6) "oth-class-taut:3:a")
1621  finally AOT_show ‹𝒜∃α φ{α} ≡ ∃α 𝒜φ{α}› .
1622qed
1623
1624
1625AOT_theorem "Act-Basic:11": ‹𝒜∀α(φ{α} ≡ ψ{α}) ≡ ∀α(𝒜φ{α} ≡ 𝒜ψ{α})›
1626proof(rule "≡I"; rule "→I")
1627  AOT_assume ‹𝒜∀α(φ{α} ≡ ψ{α})›
1628  AOT_hence ‹∀α𝒜(φ{α} ≡ ψ{α})›
1629    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] by blast
1630  AOT_hence ‹𝒜(φ{α} ≡ ψ{α})› for α using "∀E" by blast
1631  AOT_hence ‹𝒜φ{α} ≡ 𝒜ψ{α}› for α by (metis "Act-Basic:5" "≡E"(1))
1632  AOT_thus ‹∀α(𝒜φ{α} ≡ 𝒜ψ{α})› by (rule "∀I")
1633next
1634  AOT_assume ‹∀α(𝒜φ{α} ≡ 𝒜ψ{α})›
1635  AOT_hence ‹𝒜φ{α} ≡ 𝒜ψ{α}› for α using "∀E" by blast
1636  AOT_hence ‹𝒜(φ{α} ≡ ψ{α})› for α by (metis "Act-Basic:5" "≡E"(2))
1637  AOT_hence ‹∀α 𝒜(φ{α} ≡ ψ{α})› by (rule "∀I")
1638  AOT_thus ‹𝒜∀α(φ{α} ≡ ψ{α})›
1639    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
1640qed
1641
1642AOT_act_theorem "act-quant-uniq": ‹∀β(𝒜φ{β} ≡ β = α) ≡ ∀β(φ{β} ≡ β = α)›
1643proof(rule "≡I"; rule "→I")
1644  AOT_assume ‹∀β(𝒜φ{β} ≡ β = α)›
1645  AOT_hence ‹𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1646  AOT_hence ‹φ{β} ≡ β = α› for β
1647    using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1648    by metis
1649  AOT_thus ‹∀β(φ{β} ≡ β = α)› by (rule "∀I")
1650next
1651  AOT_assume ‹∀β(φ{β} ≡ β = α)›
1652  AOT_hence ‹φ{β} ≡ β = α› for β using "∀E" by blast
1653  AOT_hence ‹𝒜φ{β} ≡ β = α› for β
1654    using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1655    by metis
1656  AOT_thus ‹∀β(𝒜φ{β} ≡ β = α)› by (rule "∀I")
1657qed
1658
1659AOT_act_theorem "fund-cont-desc": ‹x = ιx(φ{x}) ≡ ∀z(φ{z} ≡ z = x)›
1660  using descriptions[axiom_inst] "act-quant-uniq" "≡E"(5) by fast
1661
1662AOT_act_theorem hintikka: ‹x = ιx(φ{x}) ≡ (φ{x} & ∀z (φ{z} → z = x))›
1663  using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" "fund-cont-desc" "≡E"(5) by blast
1664
1665
1666locale russel_axiom =
1667  fixes ψ
1668  assumes ψ_denotes_asm: "[v ⊨ ψ{κ}] ⟹ [v ⊨ κ↓]"
1669begin
1670AOT_act_theorem "russell-axiom": ‹ψ{ιx φ{x}} ≡ ∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
1671proof -
1672  AOT_have b: ‹∀x (x = ιx φ{x} ≡ (φ{x} & ∀z(φ{z} → z = x)))›
1673    using hintikka "∀I" by fast
1674  show ?thesis
1675  proof(rule "≡I"; rule "→I")
1676    AOT_assume c: ‹ψ{ιx φ{x}}›
1677    AOT_hence d: ‹ιx φ{x}↓› using ψ_denotes_asm by blast
1678    AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1679    then AOT_obtain a where a_def: ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1680    moreover AOT_have ‹a = ιx φ{x} ≡ (φ{a} & ∀z(φ{z} → z = a))› using b "∀E" by blast
1681    ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a)› using "≡E" by blast
1682    moreover AOT_have ‹ψ{a}›
1683    proof - 
1684      AOT_have 1: ‹∀x∀y(x = y → y = x)›
1685        by (simp add: "id-eq:2" "universal-cor")
1686      AOT_have ‹a = ιx φ{x} →  ιx φ{x} = a›
1687        by (rule "∀E"(1)[where τ="«ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1688           (auto simp: 1 d "universal-cor")
1689      AOT_thus ‹ψ{a}›
1690        using a_def c "rule=E" "→E" by blast
1691    qed
1692    ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a) & ψ{a}› by (rule "&I")
1693    AOT_thus ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})› by (rule "∃I")
1694  next
1695    AOT_assume ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
1696    then AOT_obtain b where g: ‹φ{b} & ∀z(φ{z} → z = b) & ψ{b}› using "instantiation"[rotated] by blast
1697    AOT_hence h: ‹b = ιx φ{x} ≡ (φ{b} & ∀z(φ{z} → z = b))› using b "∀E" by blast
1698    AOT_have ‹φ{b} & ∀z(φ{z} → z = b)› and j: ‹ψ{b}› using g "&E" by blast+
1699    AOT_hence ‹b = ιx φ{x}› using h "≡E" by blast
1700    AOT_thus ‹ψ{ιx φ{x}}› using j "rule=E" by blast
1701  qed
1702qed
1703end
1704
1705(* TODO: this nicely shows off using locales with the embedding, but maybe there is still a nicer way *)
1706(* TODO: sledgehammer tends to refer to ψ_denotes_asm in these instantiation instead of referring
1707         to cqt:5:a - should be fixed *)
1708interpretation "russell-axiom[exe,1]": russel_axiom ‹λ κ . «[Π]κ»›
1709  by standard (metis "cqt:5:a[1]"[axiom_inst, THEN "→E"] "&E"(2))
1710interpretation "russell-axiom[exe,2,1,1]": russel_axiom ‹λ κ . «[Π]κκ'»›
1711  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E")
1712interpretation "russell-axiom[exe,2,1,2]": russel_axiom ‹λ κ . «[Π]κ'κ»›
1713  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1714interpretation "russell-axiom[exe,2,2]": russel_axiom ‹λ κ . «[Π]κκ»›
1715  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1716interpretation "russell-axiom[exe,3,1,1]": russel_axiom ‹λ κ . «[Π]κκ'κ''»›
1717  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1718interpretation "russell-axiom[exe,3,1,2]": russel_axiom ‹λ κ . «[Π]κ'κκ''»›
1719  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1720interpretation "russell-axiom[exe,3,1,3]": russel_axiom ‹λ κ . «[Π]κ'κ''κ»›
1721  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1722interpretation "russell-axiom[exe,3,2,1]": russel_axiom ‹λ κ . «[Π]κκκ'»›
1723  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1724interpretation "russell-axiom[exe,3,2,2]": russel_axiom ‹λ κ . «[Π]κκ'κ»›
1725  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1726interpretation "russell-axiom[exe,3,2,3]": russel_axiom ‹λ κ . «[Π]κ'κκ»›
1727  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1728interpretation "russell-axiom[exe,3,3]": russel_axiom ‹λ κ . «[Π]κκκ»›
1729  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1730
1731interpretation "russell-axiom[enc,1]": russel_axiom ‹λ κ . «κ[Π]»›
1732  by standard (metis "cqt:5:b[1]"[axiom_inst, THEN "→E"] "&E"(2))
1733interpretation "russell-axiom[enc,2,1]": russel_axiom ‹λ κ . «κκ'[Π]»›
1734  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E")
1735interpretation "russell-axiom[enc,2,2]": russel_axiom ‹λ κ . «κ'κ[Π]»›
1736  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1737interpretation "russell-axiom[enc,2,3]": russel_axiom ‹λ κ . «κκ[Π]»›
1738  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1739interpretation "russell-axiom[enc,3,1,1]": russel_axiom ‹λ κ . «κκ'κ''[Π]»›
1740  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1741interpretation "russell-axiom[enc,3,1,2]": russel_axiom ‹λ κ . «κ'κκ''[Π]»›
1742  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1743interpretation "russell-axiom[enc,3,1,3]": russel_axiom ‹λ κ . «κ'κ''κ[Π]»›
1744  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1745interpretation "russell-axiom[enc,3,2,1]": russel_axiom ‹λ κ . «κκκ'[Π]»›
1746  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1747interpretation "russell-axiom[enc,3,2,2]": russel_axiom ‹λ κ . «κκ'κ[Π]»›
1748  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1749interpretation "russell-axiom[enc,3,2,3]": russel_axiom ‹λ κ . «κ'κκ[Π]»›
1750  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1751interpretation "russell-axiom[enc,3,3]": russel_axiom ‹λ κ . «κκκ[Π]»›
1752  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1753
1754AOT_act_theorem "1-exists:1": ‹ιx φ{x}↓ ≡ ∃!x φ{x}›
1755proof(rule "≡I"; rule "→I")
1756  AOT_assume ‹ιx φ{x}↓›
1757  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1758  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1759  AOT_hence ‹φ{a} & ∀z (φ{z} → z = a)› using hintikka "≡E" by blast
1760  AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))› by (rule "∃I")
1761  AOT_thus ‹∃!x φ{x}› using "uniqueness:1"[THEN "≡dfI"] by blast
1762next
1763  AOT_assume ‹∃!x φ{x}›
1764  AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))›
1765    using "uniqueness:1"[THEN "≡dfE"] by blast
1766  then AOT_obtain b where ‹φ{b} & ∀z (φ{z} → z = b)› using "instantiation"[rotated] by blast
1767  AOT_hence ‹b = ιx φ{x}› using hintikka "≡E" by blast
1768  AOT_thus ‹ιx φ{x}↓› by (metis "t=t-proper:2" "vdash-properties:6")
1769qed
1770
1771AOT_act_theorem "1-exists:2": ‹∃y(y=ιx φ{x}) ≡ ∃!x φ{x}›
1772  using "1-exists:1" "free-thms:1" "≡E"(6) by blast
1773
1774AOT_act_theorem "y-in:1": ‹x = ιx φ{x} → φ{x}›
1775  using "&E"(1) "→I" hintikka "≡E"(1) by blast
1776
1777AOT_act_theorem "y-in:2": ‹z = ιx φ{x} → φ{z}› using "y-in:1". (* TODO: same as above *)
1778
1779AOT_act_theorem "y-in:3": ‹ιx φ{x}↓ → φ{ιx φ{x}}›
1780proof(rule "→I")
1781  AOT_assume ‹ιx φ{x}↓›
1782  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1783  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1784  moreover AOT_have ‹φ{a}› using calculation hintikka "≡E"(1) "&E" by blast
1785  ultimately AOT_show ‹φ{ιx φ{x}}› using "rule=E" by blast
1786qed
1787
1788AOT_act_theorem "y-in:4": ‹∃y (y = ιx φ{x}) → φ{ιx φ{x}}›
1789  using "y-in:3"[THEN "→E"] "free-thms:1"[THEN "≡E"(2)] "→I" by blast
1790
1791
1792AOT_theorem "act-quant-nec": ‹∀β (𝒜φ{β} ≡ β = α) ≡ ∀β(𝒜𝒜φ{β} ≡ β = α)›
1793proof(rule "≡I"; rule "→I")
1794  AOT_assume ‹∀β (𝒜φ{β} ≡ β = α)›
1795  AOT_hence ‹𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1796  AOT_hence ‹𝒜𝒜φ{β} ≡ β = α› for β 
1797    by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1798  AOT_thus ‹∀β(𝒜𝒜φ{β} ≡ β = α)›
1799    by (rule "∀I")
1800next
1801  AOT_assume ‹∀β(𝒜𝒜φ{β} ≡ β = α)›
1802  AOT_hence ‹𝒜𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1803  AOT_hence ‹𝒜φ{β} ≡ β = α› for β
1804    by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1805  AOT_thus ‹∀β (𝒜φ{β} ≡ β = α)›
1806    by (rule "∀I")
1807qed
1808
1809AOT_theorem "equi-desc-descA:1": ‹x = ιx φ{x} ≡ x = ιx(𝒜φ{x})›
1810proof -
1811  AOT_have ‹x = ιx φ{x} ≡ ∀z (𝒜φ{z} ≡ z = x)›  using descriptions[axiom_inst] by blast
1812  also AOT_have ‹... ≡ ∀z (𝒜𝒜φ{z} ≡ z = x)›
1813  proof(rule "≡I"; rule "→I"; rule "∀I")
1814    AOT_assume ‹∀z (𝒜φ{z} ≡ z = x)›
1815    AOT_hence ‹𝒜φ{a} ≡ a = x› for a using "∀E" by blast
1816    AOT_thus ‹𝒜𝒜φ{a} ≡ a = x› for a by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1817  next
1818    AOT_assume ‹∀z (𝒜𝒜φ{z} ≡ z = x)›
1819    AOT_hence ‹𝒜𝒜φ{a} ≡ a = x› for a using "∀E" by blast
1820    AOT_thus ‹𝒜φ{a} ≡ a = x› for a  by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1821  qed
1822  also AOT_have ‹... ≡ x = ιx(𝒜φ{x})›
1823    using "Commutativity of ≡"[THEN "≡E"(1)] descriptions[axiom_inst] by fast
1824  finally show ?thesis .
1825qed
1826
1827AOT_theorem "equi-desc-descA:2": ‹ιx φ{x}↓ → ιx φ{x} = ιx(𝒜φ{x})›
1828proof(rule "→I")
1829  AOT_assume ‹ιx φ{x}↓›
1830  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1831  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1832  moreover AOT_have ‹a = ιx(𝒜φ{x})› using calculation "equi-desc-descA:1"[THEN "≡E"(1)] by blast
1833  ultimately AOT_show ‹ιx φ{x} = ιx(𝒜φ{x})› using "rule=E" by fast
1834qed
1835
1836AOT_theorem "nec-hintikka-scheme": ‹x = ιx φ{x} ≡ 𝒜φ{x} & ∀z(𝒜φ{z} → z = x)›
1837proof -
1838  AOT_have ‹x = ιx φ{x} ≡ ∀z(𝒜φ{z} ≡ z = x)› using descriptions[axiom_inst] by blast
1839  also AOT_have ‹… ≡ (𝒜φ{x} & ∀z(𝒜φ{z} → z = x))›
1840    using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" by fast
1841  finally show ?thesis.
1842qed
1843
1844AOT_theorem "equiv-desc-eq:1": ‹𝒜∀x(φ{x} ≡ ψ{x}) → ∀x (x = ιx φ{x} ≡ x = ιx ψ{x})›
1845proof(rule "→I"; rule "∀I")
1846  fix β
1847  AOT_assume ‹𝒜∀x(φ{x} ≡ ψ{x})›
1848  AOT_hence ‹𝒜(φ{x} ≡ ψ{x})› for x using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] "∀E"(2) by blast
1849  AOT_hence 0: ‹𝒜φ{x} ≡ 𝒜ψ{x}› for x by (metis "Act-Basic:5" "≡E"(1))
1850  AOT_have ‹β = ιx φ{x} ≡ 𝒜φ{β} & ∀z(𝒜φ{z} → z = β)› using "nec-hintikka-scheme" by blast
1851  also AOT_have ‹... ≡ 𝒜ψ{β} & ∀z(𝒜ψ{z} → z = β)›
1852  proof (rule "≡I"; rule "→I")
1853    AOT_assume 1: ‹𝒜φ{β} & ∀z(𝒜φ{z} → z = β)›
1854    AOT_hence ‹𝒜φ{z} → z = β› for z using "&E" "∀E" by blast
1855    AOT_hence ‹𝒜ψ{z} → z = β› for z using 0 "≡E" "→I" "→E" by metis
1856    AOT_hence ‹∀z(𝒜ψ{z} → z = β)› using "∀I" by fast
1857    moreover AOT_have ‹𝒜ψ{β}› using "&E" 0[THEN "≡E"(1)] 1 by blast
1858    ultimately AOT_show ‹𝒜ψ{β} & ∀z(𝒜ψ{z} → z = β)› using "&I" by blast
1859  next
1860    AOT_assume 1: ‹𝒜ψ{β} & ∀z(𝒜ψ{z} → z = β)›
1861    AOT_hence ‹𝒜ψ{z} → z = β› for z using "&E" "∀E" by blast
1862    AOT_hence ‹𝒜φ{z} → z = β› for z using 0 "≡E" "→I" "→E" by metis
1863    AOT_hence ‹∀z(𝒜φ{z} → z = β)› using "∀I" by fast
1864    moreover AOT_have ‹𝒜φ{β}› using "&E" 0[THEN "≡E"(2)] 1 by blast
1865    ultimately AOT_show ‹𝒜φ{β} & ∀z(𝒜φ{z} → z = β)› using "&I" by blast
1866  qed
1867  also AOT_have ‹... ≡ β = ιx ψ{x}›
1868    using "Commutativity of ≡"[THEN "≡E"(1)] "nec-hintikka-scheme" by blast
1869  finally AOT_show ‹β = ιx φ{x} ≡ β = ιx ψ{x}› .
1870qed
1871
1872AOT_theorem "equiv-desc-eq:2": ‹ιx φ{x}↓ & 𝒜∀x(φ{x} ≡ ψ{x}) → ιx φ{x} = ιx ψ{x}›
1873proof(rule "→I")
1874  AOT_assume ‹ιx φ{x}↓ & 𝒜∀x(φ{x} ≡ ψ{x})›
1875  AOT_hence 0: ‹∃y (y = ιx φ{x})› and
1876            1: ‹∀x (x = ιx φ{x} ≡ x = ιx ψ{x})›
1877    using "&E" "free-thms:1"[THEN "≡E"(1)] "equiv-desc-eq:1" "→E" by blast+
1878  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1879  moreover AOT_have ‹a = ιx ψ{x}› using calculation 1 "∀E" "≡E"(1) by fast
1880  ultimately AOT_show ‹ιx φ{x} = ιx ψ{x}›
1881    using "rule=E" by fast
1882qed
1883
1884AOT_theorem "equiv-desc-eq:3": ‹ιx φ{x}↓ & □∀x(φ{x} ≡ ψ{x}) → ιx φ{x} = ιx ψ{x}›
1885  using "→I" "equiv-desc-eq:2"[THEN "→E", OF "&I"] "&E" "nec-imp-act"[THEN "→E"] by metis
1886
1887(* Note: this is a special case of "exist-nec" *)
1888AOT_theorem "equiv-desc-eq:4": ‹ιx φ{x}↓ → □ιx φ{x}↓›
1889proof(rule "→I")
1890  AOT_assume ‹ιx φ{x}↓›
1891  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1892  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1893  AOT_thus ‹□ιx φ{x}↓›
1894    using "ex:2:a" "rule=E" by fast
1895qed
1896
1897AOT_theorem "equiv-desc-eq:5": ‹ιx φ{x}↓ → ∃y □(y = ιx φ{x})›
1898proof(rule "→I")
1899  AOT_assume ‹ιx φ{x}↓›
1900  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1901  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1902  AOT_hence ‹□(a = ιx φ{x})› by (metis "id-nec:2" "vdash-properties:10")
1903  AOT_thus ‹∃y □(y = ιx φ{x})› by (rule "∃I")
1904qed
1905
1906AOT_act_theorem "equiv-desc-eq2:1": ‹∀x (φ{x} ≡ ψ{x}) → ∀x (x = ιx φ{x} ≡ x = ιx ψ{x})›
1907  using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:1"[THEN "→E"]
1908        "RA[1]" "deduction-theorem" by blast
1909
1910AOT_act_theorem "equiv-desc-eq2:2": ‹ιx φ{x}↓ & ∀x (φ{x} ≡ ψ{x}) → ιx φ{x} = ιx ψ{x}›
1911  using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:2"[THEN "→E", OF "&I"]
1912        "RA[1]" "deduction-theorem" "&E" by metis
1913
1914context russel_axiom
1915begin
1916AOT_theorem "nec-russell-axiom": ‹ψ{ιx φ{x}} ≡ ∃x(𝒜φ{x} & ∀z(𝒜φ{z} → z = x) & ψ{x})›
1917proof -
1918  AOT_have b: ‹∀x (x = ιx φ{x} ≡ (𝒜φ{x} & ∀z(𝒜φ{z} → z = x)))›
1919    using "nec-hintikka-scheme" "∀I" by fast
1920  show ?thesis
1921  proof(rule "≡I"; rule "→I")
1922    AOT_assume c: ‹ψ{ιx φ{x}}›
1923    AOT_hence d: ‹ιx φ{x}↓› using ψ_denotes_asm by blast
1924    AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1925    then AOT_obtain a where a_def: ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1926    moreover AOT_have ‹a = ιx φ{x} ≡ (𝒜φ{a} & ∀z(𝒜φ{z} → z = a))› using b "∀E" by blast
1927    ultimately AOT_have ‹𝒜φ{a} & ∀z(𝒜φ{z} → z = a)› using "≡E" by blast
1928    moreover AOT_have ‹ψ{a}›
1929    proof - 
1930      AOT_have 1: ‹∀x∀y(x = y → y = x)›
1931        by (simp add: "id-eq:2" "universal-cor")
1932      AOT_have ‹a = ιx φ{x} →  ιx φ{x} = a›
1933        by (rule "∀E"(1)[where τ="«ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1934           (auto simp: d "universal-cor" 1)
1935      AOT_thus ‹ψ{a}›
1936        using a_def c "rule=E" "→E" by metis
1937    qed
1938    ultimately AOT_have ‹𝒜φ{a} & ∀z(𝒜φ{z} → z = a) & ψ{a}› by (rule "&I")
1939    AOT_thus ‹∃x(𝒜φ{x} & ∀z(𝒜φ{z} → z = x) & ψ{x})› by (rule "∃I")
1940  next
1941    AOT_assume ‹∃x(𝒜φ{x} & ∀z(𝒜φ{z} → z = x) & ψ{x})›
1942    then AOT_obtain b where g: ‹𝒜φ{b} & ∀z(𝒜φ{z} → z = b) & ψ{b}› using "instantiation"[rotated] by blast
1943    AOT_hence h: ‹b = ιx φ{x} ≡ (𝒜φ{b} & ∀z(𝒜φ{z} → z = b))› using b "∀E" by blast
1944    AOT_have ‹𝒜φ{b} & ∀z(𝒜φ{z} → z = b)› and j: ‹ψ{b}› using g "&E" by blast+
1945    AOT_hence ‹b = ιx φ{x}› using h "≡E" by blast
1946    AOT_thus ‹ψ{ιx φ{x}}› using j "rule=E" by blast
1947  qed
1948qed
1949end
1950
1951AOT_theorem "actual-desc:1": ‹ιx φ{x}↓ ≡ ∃!x 𝒜φ{x}›
1952proof (rule "≡I"; rule "→I")
1953  AOT_assume ‹ιx φ{x}↓›
1954  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1955  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1956  moreover AOT_have ‹a = ιx φ{x} ≡ ∀z(𝒜φ{z} ≡ z = a)›
1957    using descriptions[axiom_inst] by blast
1958  ultimately AOT_have ‹∀z(𝒜φ{z} ≡ z = a)›
1959    using "≡E" by blast
1960  AOT_hence ‹∃x∀z(𝒜φ{z} ≡ z = x)› by (rule "∃I")
1961  AOT_thus ‹∃!x 𝒜φ{x}›
1962    using "uniqueness:2"[THEN "≡E"(2)] by fast
1963next
1964  AOT_assume ‹∃!x 𝒜φ{x}›
1965  AOT_hence ‹∃x∀z(𝒜φ{z} ≡ z = x)›
1966    using "uniqueness:2"[THEN "≡E"(1)] by fast
1967  then AOT_obtain a where ‹∀z(𝒜φ{z} ≡ z = a)› using "instantiation"[rotated] by blast
1968  moreover AOT_have ‹a = ιx φ{x} ≡ ∀z(𝒜φ{z} ≡ z = a)›
1969    using descriptions[axiom_inst] by blast
1970  ultimately AOT_have ‹a = ιx φ{x}› using "≡E" by blast
1971  AOT_thus ‹ιx φ{x}↓› by (metis "t=t-proper:2" "vdash-properties:6")
1972qed
1973
1974AOT_theorem "actual-desc:2": ‹x = ιx φ{x} → 𝒜φ{x}›
1975  using "&E"(1) "contraposition:1[2]" "≡E"(1) "nec-hintikka-scheme" "reductio-aa:2" "vdash-properties:9" by blast
1976
1977AOT_theorem "actual-desc:3": ‹z = ιx φ{x} → 𝒜φ{z}›
1978  using "actual-desc:2". (* TODO: same as above *)
1979
1980AOT_theorem "actual-desc:4": ‹ιx φ{x}↓ → 𝒜φ{ιx φ{x}}›
1981proof(rule "→I")
1982  AOT_assume ‹ιx φ{x}↓›
1983  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1984  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1985  AOT_thus ‹𝒜φ{ιx φ{x}}›
1986    using "actual-desc:2" "rule=E" "→E" by fast
1987qed
1988
1989(* TODO: take another look at proof in PLM *)
1990AOT_theorem "actual-desc:5": ‹ιx φ{x} = ιx ψ{x} → 𝒜∀x(φ{x} ≡ ψ{x})›
1991proof(rule "→I")
1992  AOT_assume 0: ‹ιx φ{x} = ιx ψ{x}›
1993  AOT_hence φ_down: ‹ιx φ{x}↓› and ψ_down: ‹ιx ψ{x}↓›
1994    using "t=t-proper:1" "t=t-proper:2" "vdash-properties:6" by blast+
1995  AOT_hence ‹∃y (y = ιx φ{x})› and ‹∃y (y = ιx ψ{x})› by (metis "rule=I:1" "existential:1")+
1996  then AOT_obtain a and b where a_eq: ‹a = ιx φ{x}› and b_eq: ‹b = ιx ψ{x}›
1997    using "instantiation"[rotated] by metis
1998
1999  AOT_have ‹∀α∀β (α = β → β = α)› by (rule "∀I"; rule "∀I"; rule "id-eq:2")
2000  AOT_hence ‹∀β (ιx φ{x} = β → β = ιx φ{x})›
2001    using "∀E" φ_down by blast
2002  AOT_hence ‹ιx φ{x} = ιx ψ{x} → ιx ψ{x} = ιx φ{x}›
2003    using "∀E" ψ_down by blast
2004  AOT_hence 1: ‹ιx ψ{x} = ιx φ{x}› using 0
2005    "→E" by blast
2006
2007  AOT_have ‹𝒜φ{x} ≡ 𝒜ψ{x}› for x
2008  proof(rule "≡I"; rule "→I")
2009    AOT_assume ‹𝒜φ{x}›
2010    moreover AOT_have ‹𝒜φ{x} → x = a› for x
2011      using "nec-hintikka-scheme"[THEN "≡E"(1), OF a_eq, THEN "&E"(2)] "∀E" by blast
2012    ultimately AOT_have ‹x = a› using "→E" by blast
2013    AOT_hence ‹x = ιx φ{x}› using a_eq "rule=E" by blast
2014    AOT_hence ‹x = ιx ψ{x}› using 0 "rule=E" by blast
2015    AOT_thus ‹𝒜ψ{x}› by (metis "actual-desc:3" "vdash-properties:6")
2016  next
2017    AOT_assume ‹𝒜ψ{x}›
2018    moreover AOT_have ‹𝒜ψ{x} → x = b› for x
2019      using "nec-hintikka-scheme"[THEN "≡E"(1), OF b_eq, THEN "&E"(2)] "∀E" by blast
2020    ultimately AOT_have ‹x = b› using "→E" by blast
2021    AOT_hence ‹x = ιx ψ{x}› using b_eq "rule=E" by blast
2022    AOT_hence ‹x = ιx φ{x}› using 1 "rule=E" by blast
2023    AOT_thus ‹𝒜φ{x}› by (metis "actual-desc:3" "vdash-properties:6")
2024  qed
2025  AOT_hence ‹𝒜(φ{x} ≡ ψ{x})› for x by (metis "Act-Basic:5" "≡E"(2))
2026  AOT_hence ‹∀x 𝒜(φ{x} ≡ ψ{x})› by (rule "∀I")
2027  AOT_thus ‹𝒜∀x (φ{x} ≡ ψ{x})›
2028    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
2029qed    
2030
2031AOT_theorem "!box-desc:1": ‹∃!x □φ{x} → ∀y (y = ιx φ{x} → φ{y})›
2032proof(rule "→I")
2033  AOT_assume ‹∃!x □φ{x}›
2034  AOT_hence ζ: ‹∃x (□φ{x} & ∀z (□φ{z} → z = x))›
2035    using "uniqueness:1"[THEN "≡dfE"] by blast
2036  then AOT_obtain b where θ: ‹□φ{b} & ∀z (□φ{z} → z = b)› using "instantiation"[rotated] by blast
2037  AOT_show ‹∀y (y = ιx φ{x} → φ{y})›
2038  proof(rule GEN; rule "→I")
2039    fix y
2040    AOT_assume ‹y = ιx φ{x}›
2041    AOT_hence ‹𝒜φ{y} & ∀z (𝒜φ{z} → z = y)› using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
2042    AOT_hence ‹𝒜φ{b} → b = y› using "&E" "∀E" by blast
2043    moreover AOT_have ‹𝒜φ{b}› using θ[THEN "&E"(1)]  by (metis "nec-imp-act" "→E")
2044    ultimately AOT_have ‹b = y› using "→E" by blast
2045    moreover AOT_have ‹φ{b}› using θ[THEN "&E"(1)]  by (metis "qml:2"[axiom_inst] "→E") 
2046    ultimately AOT_show ‹φ{y}› using "rule=E" by blast
2047  qed
2048qed
2049
2050AOT_theorem "!box-desc:2": ‹∀x (φ{x} → □φ{x}) → (∃!x φ{x} → ∀y (y = ιx φ{x} → φ{y}))›
2051proof(rule "→I"; rule "→I")
2052  AOT_assume ‹∀x (φ{x} → □φ{x})›
2053  moreover AOT_assume ‹∃!x φ{x}›
2054  ultimately AOT_have ‹∃!x □φ{x}›
2055    using "nec-exist-!"[THEN "→E", THEN "→E"] by blast
2056  AOT_thus ‹∀y (y = ιx φ{x} → φ{y})›
2057    using "!box-desc:1" "→E" by blast
2058qed
2059
2060AOT_theorem "dr-alphabetic-thm": ‹ιν φ{ν}↓ → ιν φ{ν} = ιμ φ{μ}› (* TODO: vacuous *)
2061  by (simp add: "rule=I:1" "→I")
2062
2063AOT_theorem "RM:1[prem]": assumes ‹Γ ⊢□ φ → ψ› shows ‹□Γ ⊢□ □φ → □ψ›
2064proof -
2065  AOT_have ‹□Γ ⊢□ □(φ → ψ)› using "RN[prem]" assms by blast
2066  AOT_thus ‹□Γ ⊢□ □φ → □ψ› by (metis "qml:1"[axiom_inst] "→E")
2067qed
2068
2069AOT_theorem "RM:1": assumes ‹⊢□ φ → ψ› shows ‹⊢□ □φ → □ψ›
2070  using "RM:1[prem]" assms by blast
2071
2072lemmas RM = "RM:1"
2073
2074AOT_theorem "RM:2[prem]": assumes ‹Γ ⊢□ φ → ψ› shows ‹□Γ ⊢□ ◇φ → ◇ψ›
2075proof -
2076  AOT_have ‹Γ ⊢□ ¬ψ → ¬φ› using assms 
2077    by (simp add: "contraposition:1[1]")
2078  AOT_hence ‹□Γ ⊢□ □¬ψ → □¬φ› using "RM:1[prem]" by blast
2079  AOT_thus ‹□Γ ⊢□ ◇φ → ◇ψ›
2080    by (meson "≡dfE" "≡dfI" "conventions:5" "deduction-theorem" "modus-tollens:1")
2081qed
2082
2083AOT_theorem "RM:2": assumes ‹⊢□ φ → ψ› shows ‹⊢□ ◇φ → ◇ψ›
2084  using "RM:2[prem]" assms by blast
2085
2086lemmas "RM◇" = "RM:2"
2087
2088AOT_theorem "RM:3[prem]": assumes ‹Γ ⊢□ φ ≡ ψ› shows ‹□Γ ⊢□ □φ ≡ □ψ›
2089proof -
2090  AOT_have ‹Γ ⊢□ φ → ψ› and ‹Γ ⊢□ ψ → φ› using assms "≡E" "→I" by metis+
2091  AOT_hence ‹□Γ ⊢□ □φ → □ψ› and ‹□Γ ⊢□ □ψ → □φ› using "RM:1[prem]" by metis+
2092  AOT_thus ‹□Γ ⊢□ □φ ≡ □ψ›
2093    by (simp add: "≡I")
2094qed
2095
2096AOT_theorem "RM:3": assumes ‹⊢□ φ ≡ ψ› shows ‹⊢□ □φ ≡ □ψ›
2097  using "RM:3[prem]" assms by blast
2098
2099lemmas RE = "RM:3"
2100
2101AOT_theorem "RM:4[prem]": assumes ‹Γ ⊢□ φ ≡ ψ› shows ‹□Γ ⊢□ ◇φ ≡ ◇ψ›
2102proof -
2103  AOT_have ‹Γ ⊢□ φ → ψ› and ‹Γ ⊢□ ψ → φ› using assms "≡E" "→I" by metis+
2104  AOT_hence ‹□Γ ⊢□ ◇φ → ◇ψ› and ‹□Γ ⊢□ ◇ψ → ◇φ› using "RM:2[prem]" by metis+
2105  AOT_thus ‹□Γ ⊢□ ◇φ ≡ ◇ψ› by (simp add: "≡I")
2106qed
2107
2108AOT_theorem "RM:4": assumes ‹⊢□ φ ≡ ψ› shows ‹⊢□ ◇φ ≡ ◇ψ›
2109  using "RM:4[prem]" assms by blast
2110
2111lemmas "RE◇" = "RM:4"
2112
2113AOT_theorem "KBasic:1": ‹□φ → □(ψ → φ)›
2114  by (simp add: RM "pl:1"[axiom_inst])
2115
2116AOT_theorem "KBasic:2": ‹□¬φ → □(φ → ψ)›
2117  by (simp add: RM "useful-tautologies:3")
2118
2119AOT_theorem "KBasic:3": ‹□(φ & ψ) ≡ (□φ & □ψ)›
2120proof (rule "≡I"; rule "→I")
2121  AOT_assume ‹□(φ & ψ)›
2122  AOT_thus ‹□φ & □ψ›
2123    by (meson RM "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2124next
2125  AOT_have ‹□φ → □(ψ → (φ & ψ))› by (simp add: "RM:1" Adjunction)
2126  AOT_hence ‹□φ → (□ψ → □(φ & ψ))›  by (metis "Hypothetical Syllogism" "qml:1"[axiom_inst])
2127  moreover AOT_assume ‹□φ & □ψ›
2128  ultimately AOT_show ‹□(φ & ψ)›
2129    using "→E" "&E" by blast
2130qed
2131
2132AOT_theorem "KBasic:4": ‹□(φ ≡ ψ) ≡ (□(φ → ψ) & □(ψ → φ))›
2133proof -
2134  AOT_have θ: ‹□((φ → ψ) & (ψ → φ)) ≡ (□(φ → ψ) & □(ψ → φ))›
2135    by (fact "KBasic:3")
2136  AOT_modally_strict {
2137    AOT_have ‹(φ ≡ ψ) ≡ ((φ → ψ) & (ψ → φ))›
2138      by (fact "conventions:3"[THEN "≡Df"])
2139  }
2140  AOT_hence ξ: ‹□(φ ≡ ψ) ≡ □((φ → ψ) & (ψ → φ))›
2141    by (rule RE)
2142  with ξ and θ AOT_show ‹□(φ ≡ ψ) ≡ (□(φ → ψ) & □(ψ → φ))›
2143    using "≡E"(5) by blast
2144qed
2145
2146AOT_theorem "KBasic:5": ‹(□(φ → ψ) & □(ψ → φ)) → (□φ ≡ □ψ)›
2147proof -
2148  AOT_have ‹□(φ → ψ) → (□φ → □ψ)›
2149    by (fact "qml:1"[axiom_inst])
2150  moreover AOT_have ‹□(ψ → φ) → (□ψ → □φ)›
2151    by (fact "qml:1"[axiom_inst])
2152  ultimately AOT_have ‹(□(φ → ψ) & □(ψ → φ)) → ((□φ → □ψ) & (□ψ → □φ))›
2153    by (metis "&I" MP "Double Composition")
2154  moreover AOT_have ‹((□φ → □ψ) & (□ψ → □φ)) → (□φ ≡ □ψ)›
2155    using "conventions:3"[THEN "≡dfI"] "→I" by blast
2156  ultimately AOT_show ‹(□(φ → ψ) & □(ψ → φ)) → (□φ ≡ □ψ)›
2157    by (metis "Hypothetical Syllogism")
2158qed
2159
2160AOT_theorem "KBasic:6": ‹□(φ≡ ψ) → (□φ ≡ □ψ)›
2161  using "KBasic:4" "KBasic:5" "deduction-theorem" "≡E"(1) "vdash-properties:10" by blast
2162AOT_theorem "KBasic:7": ‹((□φ & □ψ) ∨ (□¬φ & □¬ψ)) → □(φ ≡ ψ)›
2163proof (rule "→I"; drule "∨E"(1); (rule "→I")?)
2164  AOT_assume ‹□φ & □ψ›
2165  AOT_hence ‹□φ› and ‹□ψ› using "&E" by blast+
2166  AOT_hence ‹□(φ → ψ)› and ‹□(ψ → φ)› using "KBasic:1" "→E" by blast+
2167  AOT_hence ‹□(φ → ψ) & □(ψ → φ)› using "&I" by blast
2168  AOT_thus ‹□(φ ≡ ψ)›  by (metis "KBasic:4" "≡E"(2))
2169next
2170  AOT_assume ‹□¬φ & □¬ψ›
2171  AOT_hence 0: ‹□(¬φ & ¬ψ)› using "KBasic:3"[THEN "≡E"(2)] by blast
2172  AOT_modally_strict {
2173    AOT_have ‹(¬φ & ¬ψ) → (φ ≡ ψ)›
2174      by (metis "&E"(1) "&E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
2175  }
2176  AOT_hence ‹□(¬φ & ¬ψ) → □(φ ≡ ψ)›
2177    by (rule RM)
2178  AOT_thus ‹□(φ ≡ ψ)› using 0 "→E" by blast
2179qed(auto)
2180
2181AOT_theorem "KBasic:8": ‹□(φ & ψ) → □(φ ≡ ψ)›
2182  by (meson "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I")
2183AOT_theorem "KBasic:9": ‹□(¬φ & ¬ψ) → □(φ ≡ ψ)›
2184  by (metis "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I" "raa-cor:4")
2185AOT_theorem "KBasic:10": ‹□φ ≡ □¬¬φ›
2186  by (simp add: "RM:3" "oth-class-taut:3:b")
2187AOT_theorem "KBasic:11": ‹¬□φ ≡ ◇¬φ›
2188proof (rule "≡I"; rule "→I")
2189  AOT_show ‹◇¬φ› if ‹¬□φ›
2190    using that "≡dfI" "conventions:5" "KBasic:10" "≡E"(3) by blast
2191next
2192  AOT_show ‹¬□φ› if ‹◇¬φ›
2193    using "≡dfE" "conventions:5" "KBasic:10" "≡E"(4) that by blast
2194qed
2195AOT_theorem "KBasic:12": ‹□φ ≡ ¬◇¬φ›
2196proof (rule "≡I"; rule "→I")
2197  AOT_show ‹¬◇¬φ› if ‹□φ›
2198    using "¬¬I" "KBasic:11" "≡E"(3) that by blast
2199next
2200  AOT_show ‹□φ› if ‹¬◇¬φ›
2201  using "KBasic:11" "≡E"(1) "reductio-aa:1" that by blast
2202qed
2203AOT_theorem "KBasic:13": ‹□(φ → ψ) → (◇φ → ◇ψ)›
2204proof -
2205  AOT_have ‹φ → ψ ⊢□ φ → ψ› by blast
2206  AOT_hence ‹□(φ → ψ) ⊢□ ◇φ → ◇ψ›
2207    using "RM:2[prem]" by blast
2208  AOT_thus ‹□(φ → ψ) → (◇φ → ◇ψ)› using "→I" by blast
2209qed
2210lemmas "K◇" = "KBasic:13"
2211AOT_theorem "KBasic:14": ‹◇□φ ≡ ¬□◇¬φ›
2212  by (meson "RE◇" "KBasic:11" "KBasic:12" "≡E"(6) "oth-class-taut:3:a")
2213AOT_theorem "KBasic:15": ‹(□φ ∨ □ψ) → □(φ ∨ ψ)›
2214proof -
2215  AOT_modally_strict {
2216    AOT_have ‹φ → (φ ∨ ψ)› and ‹ψ → (φ ∨ ψ)›
2217      by (auto simp: "Disjunction Addition"(1) "Disjunction Addition"(2))
2218  }
2219  AOT_hence ‹□φ → □(φ ∨ ψ)› and ‹□ψ → □(φ ∨ ψ)›
2220    using RM by blast+
2221  AOT_thus ‹(□φ ∨ □ψ) → □(φ ∨ ψ)›
2222    by (metis "∨E"(1) "deduction-theorem")
2223qed
2224
2225AOT_theorem "KBasic:16": ‹(□φ & ◇ψ) → ◇(φ & ψ)›
2226  by (meson "KBasic:13" "RM:1" Adjunction "Hypothetical Syllogism" Importation "vdash-properties:6")
2227
2228AOT_theorem "rule-sub-lem:1:a":
2229  assumes ‹⊢□ □(ψ ≡ χ)›
2230  shows ‹⊢□ ¬ψ ≡ ¬χ›
2231  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2232        "≡E"(1) "oth-class-taut:4:b" by blast
2233
2234AOT_theorem "rule-sub-lem:1:b":
2235  assumes ‹⊢□ □(ψ ≡ χ)›
2236  shows ‹⊢□ (ψ → Θ) ≡ (χ → Θ)›
2237  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2238  using "oth-class-taut:4:c" "vdash-properties:6" by blast
2239
2240AOT_theorem "rule-sub-lem:1:c":
2241  assumes ‹⊢□ □(ψ ≡ χ)›
2242  shows ‹⊢□ (Θ → ψ) ≡ (Θ → χ)›
2243  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2244  using "oth-class-taut:4:d" "vdash-properties:6" by blast
2245
2246AOT_theorem "rule-sub-lem:1:d":
2247  assumes ‹for arbitrary α: ⊢□ □(ψ{α} ≡ χ{α})›
2248  shows ‹⊢□ ∀α ψ{α} ≡ ∀α χ{α}›
2249proof -
2250  AOT_modally_strict {
2251    AOT_have ‹∀α (ψ{α} ≡ χ{α})›
2252      using "qml:2"[axiom_inst, THEN "→E", OF assms] "∀I" by fast
2253    AOT_hence 0: ‹ψ{α} ≡ χ{α}› for α using "∀E" by blast
2254    AOT_show ‹∀α ψ{α} ≡ ∀α χ{α}›
2255    proof (rule "≡I"; rule "→I")
2256      AOT_assume ‹∀α ψ{α}›
2257      AOT_hence ‹ψ{α}› for α using "∀E" by blast
2258      AOT_hence ‹χ{α}› for α using 0 "≡E" by blast
2259      AOT_thus ‹∀α χ{α}› by (rule "∀I")
2260    next
2261      AOT_assume ‹∀α χ{α}›
2262      AOT_hence ‹χ{α}› for α using "∀E" by blast
2263      AOT_hence ‹ψ{α}› for α using 0 "≡E" by blast
2264      AOT_thus ‹∀α ψ{α}› by (rule "∀I")
2265    qed
2266  }
2267qed
2268
2269AOT_theorem "rule-sub-lem:1:e":
2270  assumes ‹⊢□ □(ψ ≡ χ)›
2271  shows ‹⊢□ [λ ψ] ≡ [λ χ]›
2272  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2273  using "≡E"(1) "propositions-lemma:6" by blast
2274
2275AOT_theorem "rule-sub-lem:1:f":
2276  assumes ‹⊢□ □(ψ ≡ χ)›
2277  shows ‹⊢□ 𝒜ψ ≡ 𝒜χ›
2278  using "qml:2"[axiom_inst, THEN "→E", OF assms, THEN "RA[2]"]
2279  by (metis "Act-Basic:5" "≡E"(1))
2280
2281AOT_theorem "rule-sub-lem:1:g":
2282  assumes ‹⊢□ □(ψ ≡ χ)›
2283  shows ‹⊢□ □ψ ≡ □χ›
2284  using "KBasic:6" assms "vdash-properties:6" by blast
2285
2286text‹Note that instead of deriving @{text "rule-sub-lem:2"}, @{text "rule-sub-lem:3"}, @{text "rule-sub-lem:4"},
2287     and @{text "rule-sub-nec"}, we construct substitution methods instead.›
2288
2289class AOT_subst =
2290  fixes AOT_subst :: "('a ⇒ 𝗈) ⇒ bool"
2291    and AOT_subst_cond :: "'a ⇒ 'a ⇒ bool"
2292  assumes AOT_subst: "AOT_subst φ ⟹ AOT_subst_cond ψ χ ⟹ [v ⊨ «φ ψ» ≡ «φ χ»]"
2293
2294named_theorems AOT_substI
2295
2296instantiation 𝗈 :: AOT_subst
2297begin
2298
2299inductive AOT_subst_𝗈 where
2300  AOT_subst_𝗈_id[AOT_substI]: "AOT_subst_𝗈 (λφ. φ)"
2301| AOT_subst_𝗈_const[AOT_substI]: "AOT_subst_𝗈 (λφ. ψ)"
2302| AOT_subst_𝗈_not[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «¬Θ{φ}»)"
2303| AOT_subst_𝗈_imp[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 Ξ ⟹ AOT_subst_𝗈 (λ φ. «Θ{φ} → Ξ{φ}»)"
2304| AOT_subst_𝗈_lambda0[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. (AOT_lambda0 (Θ φ)))"
2305| AOT_subst_𝗈_act[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «𝒜Θ{φ}»)"
2306| AOT_subst_𝗈_box[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «□Θ{φ}»)"
2307| AOT_subst_𝗈_by_def[AOT_substI]: "(⋀ ψ . AOT_model_equiv_def (Θ ψ) (Ξ ψ)) ⟹ AOT_subst_𝗈 Ξ ⟹ AOT_subst_𝗈 Θ"
2308
2309definition AOT_subst_cond_𝗈 where "AOT_subst_cond_𝗈 ≡ λ ψ χ . ∀ v . [v ⊨ ψ ≡ χ]"
2310
2311instance
2312proof
2313  fix ψ χ :: 𝗈 and φ :: ‹𝗈 ⇒ 𝗈›
2314  assume cond: ‹AOT_subst_cond ψ χ›
2315  assume ‹AOT_subst φ›
2316  moreover AOT_have ‹⊢□ ψ ≡ χ› using cond unfolding AOT_subst_cond_𝗈_def by blast
2317  ultimately AOT_show ‹⊢□ φ{ψ} ≡ φ{χ}›
2318  proof (induct arbitrary: ψ χ)
2319    case AOT_subst_𝗈_id
2320    thus ?case using "≡E"(2) "oth-class-taut:4:b" "rule-sub-lem:1:a" by blast
2321  next
2322    case (AOT_subst_𝗈_const ψ)
2323    thus ?case by (simp add: "oth-class-taut:3:a")
2324  next
2325    case (AOT_subst_𝗈_not Θ)
2326    thus ?case by (simp add: RN "rule-sub-lem:1:a")
2327  next
2328    case (AOT_subst_𝗈_imp Θ Ξ)
2329    thus ?case by (meson RN "≡E"(5) "rule-sub-lem:1:b" "rule-sub-lem:1:c")
2330  next
2331    case (AOT_subst_𝗈_lambda0 Θ)
2332    thus ?case by (simp add: RN "rule-sub-lem:1:e")
2333  next
2334    case (AOT_subst_𝗈_act Θ)
2335    thus ?case by (simp add: RN "rule-sub-lem:1:f")
2336  next
2337    case (AOT_subst_𝗈_box Θ)
2338    thus ?case by (simp add: RN "rule-sub-lem:1:g")
2339  next
2340    case (AOT_subst_𝗈_by_def Θ Ξ)
2341    AOT_modally_strict {
2342      AOT_have ‹Ξ{ψ} ≡ Ξ{χ}› using AOT_subst_𝗈_by_def by simp
2343      AOT_thus ‹Θ{ψ} ≡ Θ{χ}›
2344        using "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ ψ] "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ χ]
2345        by (metis "≡E"(6) "oth-class-taut:3:a")
2346    }
2347  qed
2348qed
2349end
2350
2351instantiation "fun" :: (AOT_Term_id_2, AOT_subst) AOT_subst
2352begin
2353
2354definition AOT_subst_cond_fun :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b) ⇒ bool" where
2355  "AOT_subst_cond_fun ≡ λ φ ψ . ∀ α . AOT_subst_cond (φ (AOT_term_of_var α)) (ψ (AOT_term_of_var α))"
2356
2357inductive AOT_subst_fun :: "(('a ⇒ 'b) ⇒ 𝗈) ⇒ bool" where
2358  AOT_subst_fun_const[AOT_substI]: "AOT_subst_fun (λφ. ψ)"
2359| AOT_subst_fun_id[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst_fun (λφ. Ψ (φ (AOT_term_of_var α)))"
2360| AOT_subst_fun_all[AOT_substI]: "AOT_subst Ψ ⟹ (⋀ α . AOT_subst_fun (Θ (AOT_term_of_var α))) ⟹ AOT_subst_fun (λφ :: 'a ⇒ 'b. Ψ «∀α «Θ (α::'a) φ»»)"
2361| AOT_subst_fun_not[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst_fun (λφ. «¬«Ψ φ»»)"
2362| AOT_subst_fun_imp[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst Θ ⟹ AOT_subst_fun (λφ. ««Ψ φ» → «Θ φ»»)"
2363| AOT_subst_fun_lambda0[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. (AOT_lambda0 (Θ φ)))"
2364| AOT_subst_fun_act[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. «𝒜«Θ φ»»)"
2365| AOT_subst_fun_box[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. «□«Θ φ»»)"
2366| AOT_subst_fun_def[AOT_substI]: "(⋀ φ . AOT_model_equiv_def (Θ φ) (Ψ φ)) ⟹ AOT_subst_fun Ψ ⟹ AOT_subst_fun Θ"
2367
2368instance proof
2369  fix ψ χ :: "'a ⇒ 'b" and φ :: ‹('a ⇒ 'b) ⇒ 𝗈›
2370  assume ‹AOT_subst φ›
2371  moreover assume cond: ‹AOT_subst_cond ψ χ›
2372  ultimately AOT_show ‹⊢□ «φ ψ» ≡ «φ χ»›
2373  proof(induct)
2374    case (AOT_subst_fun_const ψ)
2375    then show ?case by (simp add: "oth-class-taut:3:a")
2376  next
2377  case (AOT_subst_fun_id Ψ x)
2378  then show ?case by (simp add: AOT_subst AOT_subst_cond_fun_def) 
2379  next
2380  next
2381  case (AOT_subst_fun_all Ψ Θ)
2382  AOT_have ‹⊢□ □(Θ{α, «ψ»} ≡ Θ{α, «χ»})› for α
2383    using AOT_subst_fun_all.hyps(3) AOT_subst_fun_all.prems RN by presburger
2384  thus ?case using AOT_subst[OF AOT_subst_fun_all(1)]
2385    by (simp add: RN "rule-sub-lem:1:d" AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def)
2386  next
2387  case (AOT_subst_fun_not Ψ)
2388  then show ?case by (simp add: RN "rule-sub-lem:1:a")
2389  next
2390  case (AOT_subst_fun_imp Ψ Θ)
2391  then show ?case 
2392    unfolding AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def
2393    by (meson "≡E"(5) "oth-class-taut:4:c" "oth-class-taut:4:d" "vdash-properties:6")
2394  next
2395  case (AOT_subst_fun_lambda0 Θ)
2396  then show ?case by (simp add: RN "rule-sub-lem:1:e")
2397  next
2398  case (AOT_subst_fun_act Θ)
2399  then show ?case by (simp add: RN "rule-sub-lem:1:f")
2400  next
2401  case (AOT_subst_fun_box Θ)
2402  then show ?case by (simp add: RN "rule-sub-lem:1:g")
2403  next
2404  case (AOT_subst_fun_def Θ Ψ)
2405  then show ?case
2406    by (meson "df-rules-formulas[3]" "df-rules-formulas[4]" "≡I" "≡E"(5))
2407  qed
2408qed
2409end
2410
2411ML‹
2412fun prove_AOT_subst_tac ctxt = REPEAT (SUBGOAL (fn (trm,_) => let
2413          fun findHeadConst (Const x) = SOME x
2414            | findHeadConst (A $ _) = findHeadConst A
2415            | findHeadConst _ = NONE
2416          fun findDef (Const (const_name‹AOT_model_equiv_def›, _) $ lhs $ _) = findHeadConst lhs
2417            | findDef (A $ B) = (case findDef A of SOME x => SOME x | _ => findDef B)
2418            | findDef (Abs (_,_,c)) = findDef c
2419            | findDef _ = NONE
2420          val const_opt = (findDef trm)
2421          val defs = case const_opt of SOME const => List.filter (fn thm => let
2422              val concl = Thm.concl_of thm
2423              val thmconst = (findDef concl)
2424              in case thmconst of SOME (c,_) => fst const = c | _ => false end) (AOT_Definitions.get ctxt)
2425              | _ => []
2426          val tac = case defs of [] => safe_step_tac (ctxt addSIs @{thms AOT_substI}) 1
2427                    | _ => resolve_tac ctxt defs 1
2428        in tac end) 1)
2429fun getSubstThm ctxt reversed phi p q = let
2430val p_ty = Term.type_of p
2431val abs = HOLogic.mk_Trueprop (@{const AOT_subst(_)} $ phi)
2432val abs = Syntax.check_term ctxt abs
2433val substThm = Goal.prove ctxt [] [] abs (fn {context=ctxt, prems=_} => prove_AOT_subst_tac ctxt)
2434val substThm = substThm RS @{thm AOT_subst}
2435in if reversed then let
2436  val substThm = Drule.instantiate_normalize ([],[((("χ", 0), p_ty), Thm.cterm_of ctxt p),
2437          ((("ψ", 0), p_ty), Thm.cterm_of ctxt q)]) substThm
2438  val substThm = substThm RS @{thm "≡E"(1)}
2439  in substThm end
2440else
2441  let
2442  val substThm = Drule.instantiate_normalize ([],[((("ψ", 0), p_ty), Thm.cterm_of ctxt p),
2443          ((("χ", 0), p_ty), Thm.cterm_of ctxt q)]) substThm
2444  val substThm = substThm RS @{thm "≡E"(2)}
2445  in substThm end end
2446›
2447
2448method_setup AOT_subst = ‹
2449Scan.option (Scan.lift (Args.parens (Args.$$$ "reverse"))) --
2450Scan.lift (Args.embedded_inner_syntax -- Args.embedded_inner_syntax
2451) -- Scan.option (Scan.lift (Args.$$$ "for" -- Args.colon) |-- Scan.repeat1 (Scan.lift (Args.embedded_inner_syntax) -- Scan.option (Scan.lift (Args.$$$ "::" |-- Args.embedded_inner_syntax))))
2452>> (fn ((reversed,(raw_p,raw_q)),raw_bounds) => (fn ctxt =>
2453(Method.SIMPLE_METHOD (Subgoal.FOCUS (fn {context = ctxt, params = _, prems = prems, asms = asms, concl = concl, schematics = _} =>
2454let
2455val thms = prems
2456val ctxt' = ctxt
2457val ctxt = Context_Position.set_visible false ctxt
2458val raw_bounds = case raw_bounds of SOME bounds => bounds | _ => []
2459
2460val ctxt = (fold (fn (bound, ty) => fn ctxt =>
2461  let
2462    val bound = AOT_read_term @{nonterminal τ'} ctxt bound
2463    val ty = Option.map (Syntax.read_typ ctxt) ty
2464    val ctxt = case ty of SOME ty => let
2465        val bound = Const ("_type_constraint_", Type ("fun", [ty,ty])) $ bound
2466        val bound = Syntax.check_term ctxt bound
2467      in Variable.declare_term bound ctxt end | _ => ctxt
2468  in ctxt end)) raw_bounds ctxt
2469
2470val p = AOT_read_term @{nonterminal φ'} ctxt raw_p
2471val p = Syntax.check_term ctxt p
2472val ctxt = Variable.declare_term p ctxt
2473val q = AOT_read_term @{nonterminal φ'} ctxt raw_q
2474val q = Syntax.check_term ctxt q
2475val ctxt = Variable.declare_term q ctxt
2476
2477val bounds = (map (fn (bound, _) => Syntax.check_term ctxt (AOT_read_term @{nonterminal τ'} ctxt bound))) raw_bounds
2478val p = fold (fn bound => fn p => let in Term.abs ("α", Term.type_of bound) (Term.abstract_over (bound,p)) end) bounds p
2479val p = Syntax.check_term ctxt p
2480val p_ty = Term.type_of p
2481
2482val pat = @{const Trueprop} $ (@{const AOT_model_valid_in} $ Var (("w",0), @{typ w}) $ (Var (("φ",0), Type (type_name‹fun›, [p_ty, @{typ 𝗈}])) $ p))
2483val univ = Unify.matchers (Context.Proof ctxt) [(pat, Thm.term_of concl)]
2484val univ = hd (Seq.list_of univ) (* TODO: choose? try all? filter? *)
2485val phi = the (Envir.lookup univ (("φ",0), Type (type_name‹fun›, [p_ty, @{typ 𝗈}])))
2486
2487val q = fold (fn bound => fn q => let in Term.abs ("α", Term.type_of bound) (Term.abstract_over (bound,q)) end) bounds q
2488val q = Syntax.check_term ctxt q
2489
2490(* Reparse to report bounds as fixes. *)
2491val ctxt = Context_Position.restore_visible ctxt' ctxt
2492val ctxt' = ctxt
2493fun unsource str = fst (Input.source_content (Syntax.read_input str))
2494val (_,ctxt') = Proof_Context.add_fixes (map (fn (str,_) => (Binding.make (unsource str, Position.none), NONE, Mixfix.NoSyn)) raw_bounds) ctxt'
2495val _ = (map (fn (x,_) => Syntax.check_term ctxt (AOT_read_term @{nonterminal τ'} ctxt' x))) raw_bounds
2496val _ = AOT_read_term @{nonterminal φ'} ctxt' raw_p
2497val _ = AOT_read_term @{nonterminal φ'} ctxt' raw_q
2498
2499in
2500resolve_tac ctxt [getSubstThm ctxt (case reversed of SOME x => true | _ => false) phi p q] 1
2501THEN simp_tac (ctxt addsimps [@{thm AOT_subst_cond_𝗈_def}, @{thm AOT_subst_cond_fun_def}]) 1
2502THEN (REPEAT (resolve_tac ctxt [@{thm allI}] 1))
2503THEN (TRY (resolve_tac ctxt thms 1))
2504end
2505) ctxt 1))))
2506›
2507
2508method_setup AOT_subst_def = ‹
2509Scan.option (Scan.lift (Args.parens (Args.$$$ "reverse"))) --
2510Attrib.thm
2511>> (fn (reversed,fact) => (fn ctxt =>
2512(Method.SIMPLE_METHOD (Subgoal.FOCUS (fn {context = ctxt, params = _, prems = prems, asms = asms, concl = concl, schematics = _} =>
2513let
2514val c = Thm.concl_of fact
2515val (lhs, rhs) = case c of (const‹Trueprop› $ (const‹AOT_model_equiv_def› $ lhs $ rhs)) => (lhs, rhs)
2516  | _ => raise Fail "Definition expected."
2517val substCond = HOLogic.mk_Trueprop (Const (const_name‹AOT_subst_cond›, dummyT) $ lhs $ rhs)
2518val substCond = Syntax.check_term (Proof_Context.set_mode Proof_Context.mode_schematic ctxt) substCond
2519val substCondThm = Goal.prove ctxt [] [] substCond (fn {context=ctxt, prems=prems} =>
2520      (SUBGOAL (fn (trm,int) =>
2521        auto_tac (ctxt addsimps [@{thm AOT_subst_cond_𝗈_def}, @{thm AOT_subst_cond_fun_def}, fact RS @{thm "≡Df"}])) 1))
2522val substThm = substCondThm RSN (2,@{thm AOT_subst})
2523in
2524resolve_tac ctxt [substThm RS (case reversed of NONE => @{thm "≡E"(2)} | _ => @{thm "≡E"(1)})] 1
2525THEN prove_AOT_subst_tac ctxt
2526THEN (TRY (resolve_tac ctxt prems 1))
2527end
2528) ctxt 1))))
2529›
2530
2531method_setup AOT_subst_thm = ‹
2532Scan.option (Scan.lift (Args.parens (Args.$$$ "reverse"))) --
2533Attrib.thm
2534>> (fn (reversed,fact) => (fn ctxt =>
2535(Method.SIMPLE_METHOD (Subgoal.FOCUS (fn {context = ctxt, params = _, prems = prems, asms = asms, concl = concl, schematics = _} =>
2536let
2537val c = Thm.concl_of fact
2538val (lhs, rhs) = case c of (const‹Trueprop› $ (const‹AOT_model_valid_in› $ _ $ (const‹AOT_equiv› $ lhs $ rhs))) => (lhs, rhs)
2539  | _ => raise Fail "Equivalence expected."
2540
2541val substCond = HOLogic.mk_Trueprop (Const (const_name‹AOT_subst_cond›, dummyT) $ lhs $ rhs)
2542val substCond = Syntax.check_term (Proof_Context.set_mode Proof_Context.mode_schematic ctxt) substCond
2543val substCondThm = Goal.prove ctxt [] [] substCond (fn {context=ctxt, prems=prems} =>
2544      (SUBGOAL (fn (trm,int) => auto_tac (ctxt addsimps [@{thm AOT_subst_cond_𝗈_def}, @{thm AOT_subst_cond_fun_def}, fact])) 1))
2545val substThm = substCondThm RSN (2,@{thm AOT_subst})
2546in
2547resolve_tac ctxt [substThm RS (case reversed of NONE => @{thm "≡E"(2)} | _ => @{thm "≡E"(1)})] 1
2548THEN prove_AOT_subst_tac ctxt
2549THEN (TRY (resolve_tac ctxt prems 1))
2550end
2551) ctxt 1))))
2552›
2553
2554AOT_theorem "rule-sub-remark:1[1]": assumes ‹⊢□ A!x ≡ ¬◇E!x› and ‹¬A!x› shows ‹¬¬◇E!x›
2555  by (AOT_subst (reverse) ‹¬◇E!x› ‹A!x›)
2556     (auto simp: assms) 
2557
2558AOT_theorem "rule-sub-remark:1[2]": assumes ‹⊢□ A!x ≡ ¬◇E!x› and  ‹¬¬◇E!x› shows ‹¬A!x›
2559  by (AOT_subst ‹A!x› ‹¬◇E!x›)
2560     (auto simp: assms)
2561
2562AOT_theorem "rule-sub-remark:2[1]":
2563  assumes ‹⊢□ [R]xy ≡ ([R]xy & ([Q]a ∨ ¬[Q]a))› and ‹p → [R]xy› shows ‹p → [R]xy & ([Q]a ∨ ¬[Q]a)›
2564  by (AOT_subst_thm (reverse) assms(1)) (simp add: assms(2))
2565
2566AOT_theorem "rule-sub-remark:2[2]":
2567  assumes ‹⊢□ [R]xy ≡ ([R]xy & ([Q]a ∨ ¬[Q]a))› and ‹p → [R]xy & ([Q]a ∨ ¬[Q]a)› shows ‹p → [R]xy›
2568  by (AOT_subst_thm assms(1)) (simp add: assms(2))
2569
2570AOT_theorem "rule-sub-remark:3[1]":
2571  assumes ‹for arbitrary x: ⊢□ A!x ≡ ¬◇E!x›
2572      and ‹∃x A!x›
2573    shows ‹∃x ¬◇E!x›
2574  by (AOT_subst (reverse) ‹¬◇E!x› ‹A!x› for: x)
2575     (auto simp: assms)
2576
2577AOT_theorem "rule-sub-remark:3[2]":
2578  assumes ‹for arbitrary x: ⊢□ A!x ≡ ¬◇E!x›
2579      and ‹∃x ¬◇E!x›
2580    shows ‹∃x A!x›
2581  by (AOT_subst ‹A!x› ‹¬◇E!x› for: x)
2582     (auto simp: assms)
2583
2584AOT_theorem "rule-sub-remark:4[1]":
2585  assumes ‹⊢□ ¬¬[P]x ≡ [P]x› and ‹𝒜¬¬[P]x› shows ‹𝒜[P]x›
2586  by (AOT_subst_thm (reverse) assms(1)) (simp add: assms(2))
2587
2588AOT_theorem "rule-sub-remark:4[2]":
2589  assumes ‹⊢□ ¬¬[P]x ≡ [P]x› and ‹𝒜[P]x› shows ‹𝒜¬¬[P]x›
2590  by (AOT_subst_thm assms(1)) (simp add: assms(2))
2591
2592AOT_theorem "rule-sub-remark:5[1]":
2593  assumes ‹⊢□ (φ → ψ) ≡ (¬ψ → ¬φ)› and ‹□(φ → ψ)› shows ‹□(¬ψ → ¬φ)›
2594  by (AOT_subst_thm (reverse) assms(1)) (simp add: assms(2))
2595
2596AOT_theorem "rule-sub-remark:5[2]":
2597  assumes ‹⊢□ (φ → ψ) ≡ (¬ψ → ¬φ)› and ‹□(¬ψ → ¬φ)› shows ‹□(φ → ψ)› 
2598  by (AOT_subst_thm assms(1)) (simp add: assms(2))
2599
2600AOT_theorem "rule-sub-remark:6[1]":
2601  assumes ‹⊢□ ψ ≡ χ› and ‹□(φ → ψ)› shows ‹□(φ → χ)› 
2602  by (AOT_subst_thm (reverse) assms(1)) (simp add: assms(2))
2603
2604AOT_theorem "rule-sub-remark:6[2]":
2605  assumes ‹⊢□ ψ ≡ χ› and ‹□(φ → χ)› shows ‹□(φ → ψ)›
2606  by (AOT_subst_thm assms(1)) (simp add: assms(2))
2607
2608AOT_theorem "rule-sub-remark:7[1]":
2609  assumes ‹⊢□ φ ≡ ¬¬φ› and ‹□(φ → φ)› shows ‹□(¬¬φ → φ)› 
2610  by (AOT_subst_thm (reverse) assms(1)) (simp add: assms(2))
2611
2612AOT_theorem "rule-sub-remark:7[2]":
2613  assumes ‹⊢□ φ ≡ ¬¬φ› and ‹□(¬¬φ → φ)› shows  ‹□(φ → φ)›
2614  by (AOT_subst_thm assms(1)) (simp add: assms(2))
2615
2616AOT_theorem "KBasic2:1": ‹□¬φ ≡ ¬◇φ›
2617  by (meson "conventions:5" "contraposition:2" "Hypothetical Syllogism" "df-rules-formulas[3]"
2618            "df-rules-formulas[4]" "≡I" "useful-tautologies:1")
2619
2620AOT_theorem "KBasic2:2": ‹◇(φ ∨ ψ) ≡ (◇φ ∨ ◇ψ)›
2621proof -
2622  AOT_have ‹◇(φ ∨ ψ) ≡ ◇¬(¬φ & ¬ψ)›
2623    by (simp add: "RE◇" "oth-class-taut:5:b")
2624  also AOT_have ‹… ≡ ¬□(¬φ & ¬ψ)›
2625    using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2626  also AOT_have ‹… ≡ ¬(□¬φ & □¬ψ)›
2627    using "KBasic:3" "≡E"(1) "oth-class-taut:4:b" by blast
2628  also AOT_have ‹… ≡ ¬(¬◇φ & ¬◇ψ)›
2629    using "KBasic2:1"
2630    by (AOT_subst ‹□¬φ› ‹¬◇φ›; AOT_subst ‹□¬ψ› ‹¬◇ψ›; auto simp: "oth-class-taut:3:a")
2631  also AOT_have ‹… ≡ ¬¬(◇φ ∨ ◇ψ)›
2632    using "≡E"(6) "oth-class-taut:3:b" "oth-class-taut:5:b" by blast
2633  also AOT_have ‹… ≡ ◇φ ∨ ◇ψ›
2634    by (simp add: "≡I" "useful-tautologies:1" "useful-tautologies:2")
2635  finally show ?thesis .
2636qed
2637
2638AOT_theorem "KBasic2:3": ‹◇(φ & ψ) → (◇φ & ◇ψ)›
2639  by (metis "RM◇" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "deduction-theorem" "modus-tollens:1" "reductio-aa:1")
2640
2641AOT_theorem "KBasic2:4": ‹◇(φ → ψ) ≡ (□φ → ◇ψ)›
2642proof -
2643  AOT_have ‹◇(φ → ψ) ≡ ◇(¬φ ∨ ψ)›
2644    by (AOT_subst ‹φ → ψ› ‹¬φ ∨ ψ›)
2645       (auto simp: "oth-class-taut:1:c" "oth-class-taut:3:a")
2646  also AOT_have ‹... ≡ ◇¬φ ∨ ◇ψ›
2647    by (simp add: "KBasic2:2")
2648  also AOT_have ‹... ≡ ¬□φ ∨ ◇ψ›
2649    by (AOT_subst ‹¬□φ› ‹◇¬φ›)
2650       (auto simp: "KBasic:11" "oth-class-taut:3:a")
2651  also AOT_have ‹... ≡ □φ → ◇ψ›
2652    using "≡E"(6) "oth-class-taut:1:c" "oth-class-taut:3:a" by blast
2653  finally show ?thesis .
2654qed
2655
2656AOT_theorem "KBasic2:5": ‹◇◇φ ≡ ¬□□¬φ›
2657  using "conventions:5"[THEN "≡Df"]
2658  by (AOT_subst ‹◇φ› ‹¬□¬φ›; AOT_subst ‹◇¬□¬φ› ‹¬□¬¬□¬φ›; AOT_subst (reverse) ‹¬¬□¬φ› ‹□¬φ›)
2659     (auto simp: "oth-class-taut:3:b" "oth-class-taut:3:a")
2660
2661
2662AOT_theorem "KBasic2:6": ‹□(φ ∨ ψ) → (□φ ∨ ◇ψ)›
2663proof(rule "→I"; rule "raa-cor:1")
2664  AOT_assume ‹□(φ ∨ ψ)›
2665  AOT_hence ‹□(¬φ → ψ)›
2666    using "conventions:2"[THEN "≡Df"]
2667    by (AOT_subst (reverse) ‹¬φ → ψ› ‹φ ∨ ψ›) simp
2668  AOT_hence 1: ‹◇¬φ → ◇ψ› using "KBasic:13" "vdash-properties:10" by blast
2669  AOT_assume ‹¬(□φ ∨ ◇ψ)›
2670  AOT_hence ‹¬□φ› and ‹¬◇ψ› using "&E" "≡E"(1) "oth-class-taut:5:d" by blast+
2671  AOT_thus ‹◇ψ & ¬◇ψ› using "&I"(1) 1[THEN "→E"] "KBasic:11" "≡E"(4) "raa-cor:3" by blast
2672qed
2673
2674AOT_theorem "KBasic2:7": ‹(□(φ ∨ ψ) & ◇¬φ) → ◇ψ›
2675proof(rule "→I"; frule "&E"(1); drule "&E"(2))
2676  AOT_assume ‹□(φ ∨ ψ)›
2677  AOT_hence 1: ‹□φ ∨ ◇ψ›
2678    using "KBasic2:6" "∨I"(2) "∨E"(1) by blast
2679  AOT_assume ‹◇¬φ›
2680  AOT_hence ‹¬□φ› using "KBasic:11" "≡E"(2) by blast
2681  AOT_thus ‹◇ψ› using 1 "∨E"(2) by blast
2682qed
2683
2684AOT_theorem "T-S5-fund:1": ‹φ → ◇φ›
2685  by (meson "≡dfI" "conventions:5" "contraposition:2" "Hypothetical Syllogism" "deduction-theorem" "qml:2"[axiom_inst])
2686lemmas "T◇" = "T-S5-fund:1"
2687
2688AOT_theorem "T-S5-fund:2": ‹◇□φ → □φ›
2689proof(rule "→I")
2690  AOT_assume ‹◇□φ›
2691  AOT_hence ‹¬□◇¬φ›
2692    using "KBasic:14" "≡E"(4) "raa-cor:3" by blast
2693  moreover AOT_have ‹◇¬φ → □◇¬φ›
2694    by (fact "qml:3"[axiom_inst])
2695  ultimately AOT_have ‹¬◇¬φ›
2696    using "modus-tollens:1" by blast
2697  AOT_thus ‹□φ› using "KBasic:12" "≡E"(2) by blast
2698qed
2699lemmas "5◇" = "T-S5-fund:2"
2700
2701(* Also interestingly none of these have proofs in PLM. *)
2702AOT_theorem "Act-Sub:1": ‹𝒜φ ≡ ¬𝒜¬φ›
2703  by (AOT_subst ‹𝒜¬φ› ‹¬𝒜φ›)
2704     (auto simp: "logic-actual-nec:1"[axiom_inst] "oth-class-taut:3:b")
2705
2706AOT_theorem "Act-Sub:2": ‹◇φ ≡ 𝒜◇φ›
2707  using "conventions:5"[THEN "≡Df"]
2708  by (AOT_subst ‹◇φ› ‹¬□¬φ›)
2709     (metis "deduction-theorem" "≡I" "≡E"(1) "≡E"(2) "≡E"(3)
2710            "logic-actual-nec:1"[axiom_inst] "qml-act:2"[axiom_inst])
2711
2712AOT_theorem "Act-Sub:3": ‹𝒜φ → ◇φ›
2713  using "conventions:5"[THEN "≡Df"]
2714  by (AOT_subst ‹◇φ› ‹¬□¬φ›)
2715     (metis "Act-Sub:1" "deduction-theorem" "≡E"(4) "nec-imp-act" "reductio-aa:2" "→E")
2716
2717
2718AOT_theorem "Act-Sub:4": ‹𝒜φ ≡ ◇𝒜φ›
2719proof (rule "≡I"; rule "→I")
2720  AOT_assume ‹𝒜φ›
2721  AOT_thus ‹◇𝒜φ› using "T◇" "vdash-properties:10" by blast
2722next
2723  AOT_assume ‹◇𝒜φ›
2724  AOT_hence ‹¬□¬𝒜φ›
2725    using "≡dfE" "conventions:5" by blast
2726  AOT_hence ‹¬□𝒜¬φ›
2727    by (AOT_subst ‹𝒜¬φ› ‹¬𝒜φ›)
2728       (simp add: "logic-actual-nec:1"[axiom_inst])
2729  AOT_thus ‹𝒜φ›
2730      using "Act-Basic:1" "Act-Basic:6" "∨E"(3) "≡E"(4) "reductio-aa:1" by blast
2731qed
2732
2733AOT_theorem "Act-Sub:5": ‹◇𝒜φ → 𝒜◇φ›
2734  by (metis "Act-Sub:2" "Act-Sub:3" "Act-Sub:4" "deduction-theorem" "≡E"(1) "≡E"(2) "vdash-properties:6")
2735
2736AOT_theorem "S5Basic:1": ‹◇φ ≡ □◇φ›
2737  by (simp add: "≡I" "qml:2" "qml:3" "vdash-properties:1[2]")
2738
2739AOT_theorem "S5Basic:2": ‹□φ ≡ ◇□φ›
2740  by (simp add: "T◇" "5◇" "≡I")
2741
2742AOT_theorem "S5Basic:3": ‹φ → □◇φ›
2743  using "T◇" "Hypothetical Syllogism" "qml:3" "vdash-properties:1[2]" by blast
2744lemmas "B" = "S5Basic:3"
2745
2746AOT_theorem "S5Basic:4": ‹◇□φ → φ›
2747  using "5◇" "Hypothetical Syllogism" "qml:2" "vdash-properties:1[2]" by blast
2748lemmas "B◇" = "S5Basic:4"
2749
2750AOT_theorem "S5Basic:5": ‹□φ → □□φ›
2751  using "RM:1" "B" "5◇" "Hypothetical Syllogism" by blast
2752lemmas "4" = "S5Basic:5"
2753
2754AOT_theorem "S5Basic:6": ‹□φ ≡ □□φ›
2755  by (simp add: "4" "≡I" "qml:2"[axiom_inst])
2756
2757AOT_theorem "S5Basic:7": ‹◇◇φ → ◇φ›
2758  using "conventions:5"[THEN "≡Df"] "oth-class-taut:3:b"
2759  by (AOT_subst ‹◇◇φ› ‹¬□¬◇φ›;
2760      AOT_subst ‹◇φ› ‹¬□¬φ›;
2761      AOT_subst (reverse) ‹¬¬□¬φ› ‹□¬φ›;
2762      AOT_subst (reverse) ‹□□¬φ› ‹□¬φ›)
2763     (auto simp: "S5Basic:6" "if-p-then-p")
2764
2765lemmas "4◇" = "S5Basic:7"
2766
2767AOT_theorem "S5Basic:8": ‹◇◇φ ≡ ◇φ›
2768  by (simp add: "4◇" "T◇" "≡I")
2769
2770AOT_theorem "S5Basic:9": ‹□(φ ∨ □ψ) ≡ (□φ ∨ □ψ)›
2771  apply (rule "≡I"; rule "→I")
2772  using "KBasic2:6" "5◇" "∨I"(3) "if-p-then-p" "vdash-properties:10" apply blast
2773  by (meson "KBasic:15" "4" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "con-dis-taut:7"
2774            "intro-elim:1" "Commutativity of ∨")
2775
2776AOT_theorem "S5Basic:10": ‹□(φ ∨ ◇ψ) ≡ (□φ ∨ ◇ψ)›
2777(* Note: nicely this proof is entirely sledgehammer generated *)
2778proof(rule "≡I"; rule "→I")
2779  AOT_assume ‹□(φ ∨ ◇ψ)›
2780  AOT_hence ‹□φ ∨ ◇◇ψ›
2781    by (meson "KBasic2:6" "∨I"(2) "∨E"(1))
2782  AOT_thus ‹□φ ∨ ◇ψ›
2783    by (meson "B◇" "4" "4◇" "T◇" "∨I"(3))
2784next
2785  AOT_assume ‹□φ ∨ ◇ψ›
2786  AOT_hence ‹□φ ∨ □◇ψ›
2787    by (meson "S5Basic:1" "B◇" "S5Basic:6" "T◇" "5◇" "∨I"(3) "intro-elim:1")
2788  AOT_thus ‹□(φ ∨ ◇ψ)›
2789    by (meson "KBasic:15" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
2790qed
2791
2792AOT_theorem "S5Basic:11": ‹◇(φ & ◇ψ) ≡ (◇φ & ◇ψ)›
2793proof -
2794  AOT_have ‹◇(φ & ◇ψ) ≡ ◇¬(¬φ ∨ ¬◇ψ)›
2795    by (AOT_subst ‹φ & ◇ψ› ‹¬(¬φ ∨ ¬◇ψ)›)
2796       (auto simp: "oth-class-taut:5:a" "oth-class-taut:3:a")
2797  also AOT_have ‹… ≡ ◇¬(¬φ ∨ □¬ψ)›
2798    by (AOT_subst ‹□¬ψ› ‹¬◇ψ›)
2799       (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2800  also AOT_have ‹… ≡ ¬□(¬φ ∨ □¬ψ)›
2801    using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2802  also AOT_have ‹… ≡ ¬(□¬φ ∨ □¬ψ)›
2803    using "S5Basic:9" "≡E"(1) "oth-class-taut:4:b" by blast
2804  also AOT_have ‹… ≡ ¬(¬◇φ ∨ ¬◇ψ)›
2805    using "KBasic2:1"
2806    by (AOT_subst ‹□¬φ› ‹¬◇φ›; AOT_subst ‹□¬ψ› ‹¬◇ψ›)
2807       (auto simp:  "oth-class-taut:3:a")
2808  also AOT_have ‹… ≡ ◇φ & ◇ψ›
2809    using "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:a" by blast
2810  finally show ?thesis .
2811qed
2812
2813AOT_theorem "S5Basic:12": ‹◇(φ & □ψ) ≡ (◇φ & □ψ)›
2814proof (rule "≡I"; rule "→I")
2815  AOT_assume ‹◇(φ & □ψ)›
2816  AOT_hence ‹◇φ & ◇□ψ›
2817    using "KBasic2:3" "vdash-properties:6" by blast
2818  AOT_thus ‹◇φ & □ψ›
2819    using "5◇" "&I" "&E"(1) "&E"(2) "vdash-properties:6" by blast
2820next
2821  AOT_assume ‹◇φ & □ψ›
2822  moreover AOT_have ‹(□□ψ & ◇φ) → ◇(φ & □ψ)›
2823    by (AOT_subst ‹φ & □ψ› ‹□ψ & φ›)
2824       (auto simp: "Commutativity of &" "KBasic:16")
2825  ultimately AOT_show ‹◇(φ & □ψ)›
2826    by (metis "4" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2827qed
2828
2829
2830AOT_theorem "S5Basic:13": ‹□(φ → □ψ) ≡ □(◇φ → ψ)›
2831proof (rule "≡I")
2832  AOT_modally_strict {
2833    AOT_have ‹□(φ → □ψ) → (◇φ → ψ)›
2834      by (meson "KBasic:13" "B◇" "Hypothetical Syllogism" "deduction-theorem")
2835  }
2836  AOT_hence ‹□□(φ → □ψ) → □(◇φ → ψ)›
2837    by (rule RM)
2838  AOT_thus  ‹□(φ → □ψ) → □(◇φ → ψ)›
2839    using "4" "Hypothetical Syllogism" by blast
2840next
2841  AOT_modally_strict {
2842    AOT_have ‹□(◇φ → ψ) → (φ → □ψ)›
2843      by (meson "B" "Hypothetical Syllogism" "deduction-theorem" "qml:1" "vdash-properties:1[2]")
2844  }
2845  AOT_hence  ‹□□(◇φ → ψ) → □(φ → □ψ)›
2846    by (rule RM)
2847  AOT_thus ‹□(◇φ → ψ) → □(φ → □ψ)›
2848    using "4" "Hypothetical Syllogism" by blast
2849qed
2850
2851AOT_theorem "derived-S5-rules:1":
2852  assumes ‹Γ ⊢□ ◇φ → ψ› shows ‹□Γ ⊢□ φ → □ψ›
2853proof -
2854  AOT_have ‹□Γ ⊢□ □◇φ → □ψ›
2855    using assms by (rule "RM:1[prem]")
2856  AOT_thus ‹□Γ ⊢□ φ → □ψ›
2857    using "B" "Hypothetical Syllogism" by blast
2858qed
2859
2860AOT_theorem "derived-S5-rules:2":
2861  assumes ‹Γ ⊢□ φ → □ψ› shows ‹□Γ ⊢□ ◇φ → ψ›
2862proof -
2863  AOT_have ‹□Γ ⊢□ ◇φ → ◇□ψ›
2864    using assms by (rule "RM:2[prem]")
2865  AOT_thus ‹□Γ ⊢□ ◇φ → ψ›
2866    using "B◇" "Hypothetical Syllogism" by blast
2867qed
2868
2869AOT_theorem "BFs:1": ‹∀α □φ{α} → □∀α φ{α}›
2870proof -
2871  AOT_modally_strict {
2872    AOT_modally_strict {
2873      AOT_have ‹∀α □φ{α} → □φ{α}› for α by (fact AOT)
2874    }
2875    AOT_hence ‹◇∀α □φ{α} → ◇□φ{α}› for α by (rule "RM◇")
2876    AOT_hence ‹◇∀α □φ{α} → ∀α φ{α}›
2877      using "B◇" "∀I" "→E" "→I" by metis
2878  }
2879  thus ?thesis using "derived-S5-rules:1" by blast
2880qed
2881lemmas "BF" = "BFs:1"
2882
2883AOT_theorem "BFs:2": ‹□∀α φ{α} → ∀α □φ{α}›
2884proof -
2885  AOT_have ‹□∀α φ{α} → □φ{α}› for α using RM "cqt-orig:3" by metis
2886  thus ?thesis using  "cqt-orig:2"[THEN "→E"] "∀I" by metis
2887qed
2888lemmas "CBF" = "BFs:2"
2889
2890AOT_theorem "BFs:3": ‹◇∃α φ{α} → ∃α ◇φ{α}›
2891proof(rule "→I")
2892  AOT_modally_strict {
2893    AOT_have ‹□∀α ¬φ{α} ≡ ∀α □¬φ{α}›
2894      using BF CBF "≡I" by blast
2895  } note θ = this
2896
2897  AOT_assume ‹◇∃α φ{α}›
2898  AOT_hence ‹¬□¬(∃α φ{α})›
2899    using "≡dfE" "conventions:5" by blast
2900  AOT_hence ‹¬□∀α ¬φ{α}›
2901    apply (AOT_subst ‹∀α ¬φ{α}› ‹¬(∃α φ{α})›)
2902    using "≡dfI" "conventions:3" "conventions:4" "&I" "contraposition:2" "cqt-further:4"
2903          "df-rules-formulas[1]" "vdash-properties:1[2]" by blast
2904  AOT_hence ‹¬∀α □¬φ{α}›
2905    apply (AOT_subst (reverse) ‹∀α □¬φ{α}› ‹□∀α ¬φ{α}›)
2906    using θ by blast
2907  AOT_hence ‹¬∀α ¬¬□¬φ{α}›
2908    by (AOT_subst (reverse) ‹¬¬□¬φ{α}› ‹□¬φ{α}› for: α)
2909       (simp add: "oth-class-taut:3:b")
2910  AOT_hence ‹∃α ¬□¬φ{α}›
2911    by (rule "conventions:4"[THEN "≡dfI"])
2912  AOT_thus ‹∃α ◇φ{α}›
2913    using "conventions:5"[THEN "≡Df"]
2914    by (AOT_subst ‹◇φ{α}› ‹¬□¬φ{α}› for: α)
2915qed
2916lemmas "BF◇" = "BFs:3"
2917
2918AOT_theorem "BFs:4": ‹∃α ◇φ{α} → ◇∃α φ{α}›
2919proof(rule "→I")
2920  AOT_assume ‹∃α ◇φ{α}›
2921  AOT_hence ‹¬∀α ¬◇φ{α}›
2922    using "conventions:4"[THEN "≡dfE"] by blast
2923  AOT_hence ‹¬∀α □¬φ{α}›
2924    using "KBasic2:1"
2925    by (AOT_subst ‹□¬φ{α}› ‹¬◇φ{α}› for: α)
2926  moreover AOT_have ‹∀α □¬φ{α} ≡ □∀α ¬φ{α}›
2927    using "≡I" "BF" "CBF" by metis
2928  ultimately AOT_have 1: ‹¬□∀α ¬φ{α}›
2929    using "≡E"(3) by blast
2930  AOT_show ‹◇∃α φ{α}›
2931    apply (rule "conventions:5"[THEN "≡dfI"])
2932    apply (AOT_subst ‹∃α φ{α}› ‹¬∀α ¬φ{α}›)
2933     apply (simp add: "conventions:4" "≡Df")
2934    apply (AOT_subst ‹¬¬∀α ¬φ{α}› ‹∀α ¬φ{α}›)
2935    by (auto simp: 1 "≡I" "useful-tautologies:1" "useful-tautologies:2")
2936qed
2937lemmas "CBF◇" = "BFs:4"
2938
2939AOT_theorem "sign-S5-thm:1": ‹∃α □φ{α} → □∃α φ{α}›
2940proof(rule "→I")
2941  AOT_assume ‹∃α □φ{α}›
2942  then AOT_obtain α where ‹□φ{α}› using "∃E" by metis
2943  moreover AOT_have ‹□α↓›
2944    by (simp add: "ex:1:a" "rule-ui:2[const_var]" RN)
2945  moreover AOT_have ‹□φ{τ}, □τ↓ ⊢□ □∃α φ{α}› for τ
2946  proof -
2947    AOT_have ‹φ{τ}, τ↓ ⊢□ ∃α φ{α}› using "existential:1" by blast
2948    AOT_thus ‹□φ{τ}, □τ↓ ⊢□ □∃α φ{α}›
2949      using "RN[prem]"[where Γ="{φ τ, «τ↓»}", simplified] by blast
2950  qed
2951  ultimately AOT_show ‹□∃α φ{α}› by blast
2952qed
2953lemmas Buridan = "sign-S5-thm:1"
2954
2955AOT_theorem "sign-S5-thm:2": ‹◇∀α φ{α} → ∀α ◇φ{α}›
2956proof -
2957  AOT_have ‹∀α (◇∀α φ{α} → ◇φ{α})›
2958    by (simp add: "RM◇" "cqt-orig:3" "∀I")
2959  AOT_thus ‹◇∀α φ{α} → ∀α ◇φ{α}›
2960    using "∀E"(4) "∀I" "→E" "→I" by metis
2961qed
2962lemmas "Buridan◇" = "sign-S5-thm:2"
2963
2964AOT_theorem "sign-S5-thm:3": ‹◇∃α (φ{α} & ψ{α}) → ◇(∃α φ{α} & ∃α ψ{α})›
2965  apply (rule "RM:2")
2966  by (metis (no_types, lifting) "instantiation" "&I" "&E"(1)
2967                                "&E"(2) "deduction-theorem" "existential:2[const_var]")
2968
2969AOT_theorem "sign-S5-thm:4": ‹◇∃α (φ{α} & ψ{α}) → ◇∃α φ{α}›
2970  apply (rule "RM:2")
2971  by (meson "instantiation" "&E"(1) "deduction-theorem" "existential:2[const_var]")
2972
2973AOT_theorem "sign-S5-thm:5": ‹(□∀α (φ{α} → ψ{α}) & □∀α (ψ{α} → χ{α})) → □∀α (φ{α} → χ{α})›
2974proof -
2975  {
2976    fix φ' ψ' χ'
2977    AOT_assume ‹⊢□ φ' & ψ' → χ'›
2978    AOT_hence ‹□φ' & □ψ' → □χ'›
2979      using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2980      using "&E" "&I" "→E" "→I" by metis
2981  } note R = this
2982  show ?thesis by (rule R; fact AOT)
2983qed
2984
2985AOT_theorem "sign-S5-thm:6": ‹(□∀α (φ{α} ≡ ψ{α}) & □∀α(ψ{α} ≡ χ{α})) → □∀α(φ{α} ≡ χ{α})›
2986proof -
2987  {
2988    fix φ' ψ' χ'
2989    AOT_assume ‹⊢□ φ' & ψ' → χ'›
2990    AOT_hence ‹□φ' & □ψ' → □χ'›
2991      using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2992      using "&E" "&I" "→E" "→I" by metis
2993  } note R = this
2994  show ?thesis by (rule R; fact AOT)
2995qed
2996
2997AOT_theorem "exist-nec2:1": ‹◇τ↓ → τ↓›
2998  using "B◇" "RM◇" "Hypothetical Syllogism" "exist-nec" by blast
2999
3000AOT_theorem "exists-nec2:2": ‹◇τ↓ ≡ □τ↓›
3001  by (meson "Act-Sub:3" "Hypothetical Syllogism" "exist-nec" "exist-nec2:1" "≡I" "nec-imp-act")
3002
3003AOT_theorem "exists-nec2:3": ‹¬τ↓ → □¬τ↓›
3004  using "KBasic2:1" "deduction-theorem" "exist-nec2:1" "≡E"(2) "modus-tollens:1" by blast
3005
3006AOT_theorem "exists-nec2:4": ‹◇¬τ↓ ≡ □¬τ↓›
3007  by (metis "Act-Sub:3" "KBasic:12" "deduction-theorem" "exist-nec" "exists-nec2:3" "≡I" "≡E"(4) "nec-imp-act" "reductio-aa:1")
3008
3009AOT_theorem "id-nec2:1": ‹◇α = β → α = β›
3010  using "B◇" "RM◇" "Hypothetical Syllogism" "id-nec:1" by blast
3011
3012AOT_theorem "id-nec2:2": ‹α ≠ β → □α ≠ β›
3013  apply (AOT_subst ‹α ≠ β› ‹¬(α = β)›)
3014  using "=-infix"[THEN "≡Df"] apply blast
3015  using "KBasic2:1" "deduction-theorem" "id-nec2:1" "≡E"(2) "modus-tollens:1" by blast
3016
3017AOT_theorem "id-nec2:3": ‹◇α ≠ β → α ≠ β›
3018  apply (AOT_subst ‹α ≠ β› ‹¬(α = β)›)
3019  using "=-infix"[THEN "≡Df"] apply blast
3020  by (metis "KBasic:11" "deduction-theorem" "id-nec:2" "≡E"(3) "reductio-aa:2" "vdash-properties:6")
3021
3022AOT_theorem "id-nec2:4": ‹◇α = β → □α = β›
3023  using "Hypothetical Syllogism" "id-nec2:1" "id-nec:1" by blast
3024
3025AOT_theorem "id-nec2:5": ‹◇α ≠ β → □α ≠ β›
3026  using "id-nec2:3" "id-nec2:2" "→I" "→E" by metis
3027
3028AOT_theorem "sc-eq-box-box:1": ‹□(φ → □φ) ≡ (◇φ → □φ)›
3029  apply (rule "≡I"; rule "→I")
3030  using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:10" apply blast
3031  by (metis "KBasic2:1" "KBasic:1" "KBasic:2" "S5Basic:13" "≡E"(2) "raa-cor:5" "vdash-properties:6")
3032
3033AOT_theorem "sc-eq-box-box:2": ‹(□(φ → □φ) ∨ (◇φ → □φ)) → (◇φ ≡ □φ)›
3034  by (metis "Act-Sub:3" "KBasic:13" "5◇" "∨E"(2) "deduction-theorem" "≡I" "nec-imp-act" "raa-cor:2" "vdash-properties:10")
3035
3036AOT_theorem "sc-eq-box-box:3": ‹□(φ → □φ) → (¬□φ ≡ □¬φ)›
3037proof (rule "→I"; rule "≡I"; rule "→I")
3038  AOT_assume ‹□(φ → □φ)›
3039  AOT_hence ‹◇φ → □φ› using "sc-eq-box-box:1" "≡E" by blast
3040  moreover AOT_assume ‹¬□φ›
3041  ultimately AOT_have ‹¬◇φ›
3042    using "modus-tollens:1" by blast
3043  AOT_thus ‹□¬φ›
3044    using "KBasic2:1" "≡E"(2) by blast
3045next
3046  AOT_assume ‹□(φ → □φ)›
3047  moreover AOT_assume ‹□¬φ›
3048  ultimately AOT_show ‹¬□φ›
3049    using "modus-tollens:1" "qml:2" "vdash-properties:10" "vdash-properties:1[2]" by blast
3050qed
3051
3052AOT_theorem "sc-eq-box-box:4": ‹(□(φ → □φ) & □(ψ → □ψ)) → ((□φ ≡ □ψ) → □(φ ≡ ψ))›
3053proof(rule "→I"; rule "→I")
3054  AOT_assume θ: ‹□(φ → □φ) & □(ψ → □ψ)›
3055  AOT_assume ξ: ‹□φ ≡ □ψ›
3056  AOT_hence ‹(□φ & □ψ) ∨ (¬□φ & ¬□ψ)›
3057    using "≡E"(4) "oth-class-taut:4:g" "raa-cor:3" by blast
3058  moreover {
3059    AOT_assume ‹□φ & □ψ›
3060    AOT_hence ‹□(φ ≡ ψ)›
3061      using "KBasic:3" "KBasic:8" "≡E"(2) "vdash-properties:10" by blast
3062  }
3063  moreover {
3064    AOT_assume ‹¬□φ & ¬□ψ›
3065    moreover AOT_have ‹¬□φ ≡ □¬φ› and ‹¬□ψ ≡ □¬ψ›
3066      using θ "Conjunction Simplification"(1) "Conjunction Simplification"(2) "sc-eq-box-box:3" "vdash-properties:10" by metis+
3067    ultimately AOT_have ‹□¬φ & □¬ψ›
3068      by (metis "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "≡E"(4) "modus-tollens:1" "raa-cor:3")
3069    AOT_hence ‹□(φ ≡ ψ)›
3070      using "KBasic:3" "KBasic:9" "≡E"(2) "vdash-properties:10" by blast
3071  }
3072  ultimately AOT_show ‹□(φ ≡ ψ)›
3073    using "∨E"(2) "reductio-aa:1" by blast
3074qed
3075
3076AOT_theorem "sc-eq-box-box:5": ‹(□(φ → □φ) & □(ψ → □ψ)) → □((φ ≡ ψ) → □(φ ≡ ψ))›
3077proof (rule "→I")
3078  AOT_assume ‹(□(φ → □φ) & □(ψ → □ψ))›
3079  AOT_hence ‹□(□(φ → □φ) & □(ψ → □ψ))›
3080    using 4[THEN "→E"] "&E" "&I" "KBasic:3" "≡E"(2) by metis
3081  moreover AOT_have ‹□(□(φ → □φ) & □(ψ → □ψ)) → □((φ ≡ ψ) → □(φ ≡ ψ))›
3082  proof (rule RM; rule "→I"; rule "→I")
3083    AOT_modally_strict {
3084      AOT_assume A: ‹(□(φ → □φ) & □(ψ → □ψ))›
3085      AOT_hence ‹φ → □φ› and ‹ψ → □ψ›
3086        using "&E" "qml:2"[axiom_inst] "→E" by blast+
3087      moreover AOT_assume ‹φ ≡ ψ›
3088      ultimately AOT_have ‹□φ ≡ □ψ›
3089        using "→E" "qml:2"[axiom_inst] "≡E" "≡I" by meson
3090      moreover AOT_have ‹(□φ ≡ □ψ) → □(φ ≡ ψ)›
3091        using A "sc-eq-box-box:4" "→E" by blast
3092      ultimately AOT_show ‹□(φ ≡ ψ)› using "→E" by blast
3093    }
3094  qed
3095  ultimately AOT_show ‹□((φ ≡ ψ) → □(φ ≡ ψ))› using "→E" by blast
3096qed
3097
3098AOT_theorem "sc-eq-box-box:6": ‹□(φ → □φ) → ((φ → □ψ) → □(φ → ψ))›
3099proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3100  AOT_assume ‹¬□(φ → ψ)›
3101  AOT_hence ‹◇¬(φ → ψ)› by (metis "KBasic:11" "≡E"(1))
3102  AOT_hence ‹◇(φ & ¬ψ)›
3103    by (AOT_subst ‹φ & ¬ψ› ‹¬(φ → ψ)›)
3104       (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3105  AOT_hence ‹◇φ› and 2: ‹◇¬ψ› using "KBasic2:3"[THEN "→E"] "&E" by blast+
3106  moreover AOT_assume ‹□(φ → □φ)›
3107  ultimately AOT_have ‹□φ› by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3108  AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3109  moreover AOT_assume ‹φ → □ψ›
3110  ultimately AOT_have ‹□ψ› using "→E" by blast
3111  moreover AOT_have ‹¬□ψ› using 2 "KBasic:12" "¬¬I" "intro-elim:3:d" by blast
3112  ultimately AOT_show ‹□ψ & ¬□ψ› using "&I" by blast
3113qed
3114
3115AOT_theorem "sc-eq-box-box:7": ‹□(φ → □φ) → ((φ → 𝒜ψ) → 𝒜(φ → ψ))›
3116proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3117  AOT_assume ‹¬𝒜(φ → ψ)›
3118  AOT_hence ‹𝒜¬(φ → ψ)› by (metis "Act-Basic:1" "∨E"(2))
3119  AOT_hence ‹𝒜(φ & ¬ψ)›
3120    by (AOT_subst ‹φ & ¬ψ› ‹¬(φ → ψ)›)
3121       (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3122  AOT_hence ‹𝒜φ› and 2: ‹𝒜¬ψ› using "Act-Basic:2"[THEN "≡E"(1)] "&E" by blast+
3123  AOT_hence ‹◇φ› by (metis "Act-Sub:3" "→E")
3124  moreover AOT_assume ‹□(φ → □φ)›
3125  ultimately AOT_have ‹□φ› by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3126  AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3127  moreover AOT_assume ‹φ → 𝒜ψ›
3128  ultimately AOT_have ‹𝒜ψ› using "→E" by blast
3129  moreover AOT_have ‹¬𝒜ψ› using 2 by (meson "Act-Sub:1" "≡E"(4) "raa-cor:3")
3130  ultimately AOT_show ‹𝒜ψ & ¬𝒜ψ› using "&I" by blast
3131qed
3132
3133AOT_theorem "sc-eq-fur:1": ‹◇𝒜φ ≡ □𝒜φ›
3134  using "Act-Basic:6" "Act-Sub:4" "≡E"(6) by blast
3135
3136AOT_theorem "sc-eq-fur:2": ‹□(φ → □φ) → (𝒜φ ≡ φ)›
3137  by (metis "B◇" "Act-Sub:3" "KBasic:13" "T◇" "Hypothetical Syllogism" "deduction-theorem" "≡I" "nec-imp-act")
3138
3139AOT_theorem "sc-eq-fur:3": ‹□∀x (φ{x} → □φ{x}) → (∃!x φ{x} → ιx φ{x}↓)›
3140proof (rule "→I"; rule "→I")
3141  AOT_assume ‹□∀x (φ{x} → □φ{x})›
3142  AOT_hence A: ‹∀x □(φ{x} → □φ{x})› using CBF "→E" by blast
3143  AOT_assume ‹∃!x φ{x}›
3144  then AOT_obtain a where a_def: ‹φ{a} & ∀y (φ{y} → y = a)›
3145    using "∃E"[rotated 1, OF "uniqueness:1"[THEN "≡dfE"]] by blast
3146  moreover AOT_have ‹□φ{a}› using calculation A "∀E"(2) "qml:2"[axiom_inst] "→E" "&E"(1) by blast
3147  AOT_hence ‹𝒜φ{a}› using "nec-imp-act" "vdash-properties:6" by blast
3148  moreover AOT_have ‹∀y (𝒜φ{y} → y = a)›
3149  proof (rule "∀I"; rule "→I")
3150    fix b
3151    AOT_assume ‹𝒜φ{b}›
3152    AOT_hence ‹◇φ{b}›
3153      using "Act-Sub:3" "vdash-properties:6" by blast
3154    moreover {
3155      AOT_have ‹□(φ{b} → □φ{b})›
3156        using A "∀E"(2) by blast
3157      AOT_hence ‹◇φ{b} → □φ{b}›
3158        using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:6" by blast
3159    }
3160    ultimately AOT_have ‹□φ{b}› using "→E" by blast
3161    AOT_hence ‹φ{b}› using "qml:2"[axiom_inst] "→E" by blast
3162    AOT_thus ‹b = a›
3163      using a_def[THEN "&E"(2)] "∀E"(2) "→E" by blast
3164  qed
3165  ultimately AOT_have ‹𝒜φ{a} & ∀y (𝒜φ{y} → y = a)›
3166    using "&I" by blast
3167  AOT_hence ‹∃x (𝒜φ{x} & ∀y (𝒜φ{y} → y = x))› using "∃I" by fast
3168  AOT_hence ‹∃!x 𝒜φ{x}› using "uniqueness:1"[THEN "≡dfI"] by fast
3169  AOT_thus ‹ιx φ{x}↓›
3170    using "actual-desc:1"[THEN "≡E"(2)] by blast
3171qed
3172
3173AOT_theorem "sc-eq-fur:4": ‹□∀x (φ{x} → □φ{x}) → (x = ιx φ{x} ≡ (φ{x} & ∀z (φ{z} → z = x)))›
3174proof (rule "→I")
3175  AOT_assume ‹□∀x (φ{x} → □φ{x})›
3176  AOT_hence ‹∀x □(φ{x} → □φ{x})› using CBF "→E" by blast
3177  AOT_hence A: ‹𝒜φ{α} ≡ φ{α}› for α using "sc-eq-fur:2" "∀E" "→E" by fast
3178  AOT_show ‹x = ιx φ{x} ≡ (φ{x} & ∀z (φ{z} → z = x))›
3179  proof (rule "≡I"; rule "→I")
3180    AOT_assume ‹x = ιx φ{x}›
3181    AOT_hence B: ‹𝒜φ{x} & ∀z (𝒜φ{z} → z = x)›
3182      using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
3183    AOT_show ‹φ{x} & ∀z (φ{z} → z = x)›
3184    proof (rule "&I"; (rule "∀I"; rule "→I")?)
3185      AOT_show ‹φ{x}› using A B[THEN "&E"(1)] "≡E"(1) by blast
3186    next
3187      AOT_show ‹z = x› if ‹φ{z}› for z
3188        using that B[THEN "&E"(2)] "∀E"(2) "→E" A[THEN "≡E"(2)] by blast
3189    qed
3190  next
3191    AOT_assume B: ‹φ{x} & ∀z (φ{z} → z = x)›
3192    AOT_have ‹𝒜φ{x} & ∀z (𝒜φ{z} → z = x)›
3193    proof(rule "&I"; (rule "∀I"; rule "→I")?)
3194      AOT_show ‹𝒜φ{x}› using B[THEN "&E"(1)] A[THEN "≡E"(2)] by blast
3195    next
3196      AOT_show ‹b = x› if ‹𝒜φ{b}› for b
3197        using that A[THEN "≡E"(1)] B[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] by blast
3198    qed
3199    AOT_thus ‹x = ιx φ{x}›
3200      using "nec-hintikka-scheme"[THEN "≡E"(2)] by blast
3201  qed
3202qed
3203
3204AOT_theorem "id-act:1": ‹α = β ≡ 𝒜α = β›
3205  by (meson "Act-Sub:3" "Hypothetical Syllogism" "id-nec2:1" "id-nec:2" "≡I" "nec-imp-act")
3206
3207AOT_theorem "id-act:2": ‹α ≠ β ≡ 𝒜α ≠ β›
3208proof (AOT_subst ‹α ≠ β› ‹¬(α = β)›)
3209  AOT_modally_strict {
3210    AOT_show ‹α ≠ β ≡ ¬(α = β)›
3211      by (simp add: "=-infix" "≡Df")
3212  }
3213next
3214  AOT_show ‹¬(α = β) ≡ 𝒜¬(α = β)›
3215  proof (safe intro!: "≡I" "→I")
3216    AOT_assume ‹¬α = β›
3217    AOT_hence ‹¬𝒜α = β› using "id-act:1" "≡E"(3) by blast
3218    AOT_thus ‹𝒜¬α = β›
3219      using "¬¬E" "Act-Sub:1" "≡E"(3) by blast
3220  next
3221    AOT_assume ‹𝒜¬α = β›
3222    AOT_hence ‹¬𝒜α = β›
3223      using "¬¬I" "Act-Sub:1" "≡E"(4) by blast
3224    AOT_thus ‹¬α = β›
3225      using "id-act:1" "≡E"(4) by blast
3226  qed
3227qed
3228
3229AOT_theorem "A-Exists:1": ‹𝒜∃!α φ{α} ≡ ∃!α 𝒜φ{α}›
3230proof -
3231  AOT_have ‹𝒜∃!α φ{α} ≡ 𝒜∃α∀β (φ{β} ≡ β = α)›
3232    by (AOT_subst ‹∃!α φ{α}› ‹∃α∀β (φ{β} ≡ β = α)›)
3233       (auto simp add: "oth-class-taut:3:a" "uniqueness:2")
3234  also AOT_have ‹… ≡ ∃α 𝒜∀β (φ{β} ≡ β = α)›
3235    by (simp add: "Act-Basic:10")
3236  also AOT_have ‹… ≡ ∃α∀β 𝒜(φ{β} ≡ β = α)›
3237    by (AOT_subst ‹𝒜∀β (φ{β} ≡ β = α)› ‹∀β 𝒜(φ{β} ≡ β = α)› for: α)
3238       (auto simp: "logic-actual-nec:3" "vdash-properties:1[2]" "oth-class-taut:3:a")
3239  also AOT_have ‹… ≡ ∃α∀β (𝒜φ{β} ≡ 𝒜β = α)›
3240    by (AOT_subst (reverse) ‹𝒜φ{β} ≡ 𝒜β = α›  ‹𝒜(φ{β} ≡ β = α)› for: α β :: 'a)
3241       (auto simp: "Act-Basic:5" "cqt-further:7")
3242  also AOT_have ‹… ≡ ∃α∀β (𝒜φ{β} ≡ β = α)›
3243    by (AOT_subst (reverse) ‹𝒜β = α› ‹β = α› for: α β :: 'a)
3244       (auto simp: "id-act:1" "cqt-further:7")
3245  also AOT_have ‹... ≡ ∃!α 𝒜φ{α}›
3246    using "uniqueness:2" "Commutativity of ≡"[THEN "≡E"(1)] by fast
3247  finally show ?thesis .
3248qed
3249
3250AOT_theorem "A-Exists:2": ‹ιx φ{x}↓ ≡ 𝒜∃!x φ{x}›
3251  by (AOT_subst ‹𝒜∃!x φ{x}› ‹∃!x 𝒜φ{x}›)
3252     (auto simp: "actual-desc:1" "A-Exists:1")
3253
3254AOT_theorem "id-act-desc:1": ‹ιx (x = y)↓›
3255proof(rule "existence:1"[THEN "≡dfI"]; rule "∃I")
3256  AOT_show ‹[λx E!x → E!x]ιx (x = y)›
3257  proof (rule "russell-axiom[exe,1].nec-russell-axiom"[THEN "≡E"(2)]; rule "∃I"; (rule "&I")+)
3258    AOT_show ‹𝒜y = y› by (simp add: "RA[2]" "id-eq:1")
3259  next
3260    AOT_show ‹∀z (𝒜z = y → z = y)›
3261      apply (rule "∀I")
3262      using "id-act:1"[THEN "≡E"(2)] "→I" by blast
3263  next
3264    AOT_show ‹[λx E!x → E!x]y›
3265    proof (rule "lambda-predicates:2"[axiom_inst, THEN "→E", THEN "≡E"(2)])
3266      AOT_show ‹[λx E!x → E!x]↓›
3267        by "cqt:2[lambda]"
3268    next
3269      AOT_show ‹E!y → E!y› 
3270        by (simp add: "if-p-then-p")
3271    qed
3272  qed
3273next
3274  AOT_show ‹[λx E!x → E!x]↓›
3275    by "cqt:2[lambda]"
3276qed
3277
3278AOT_theorem "id-act-desc:2": ‹y = ιx (x = y)›
3279  by (rule descriptions[axiom_inst, THEN "≡E"(2)]; rule "∀I"; rule "id-act:1"[symmetric])
3280
3281AOT_theorem "pre-en-eq:1[1]": ‹x1[F] → □x1[F]›
3282  by (simp add: encoding "vdash-properties:1[2]")
3283
3284AOT_theorem "pre-en-eq:1[2]": ‹x1x2[F] → □x1x2[F]›
3285proof (rule "→I")
3286  AOT_assume ‹x1x2[F]›
3287  AOT_hence ‹x1[λy [F]yx2]› and ‹x2[λy [F]x1y]›
3288    using "nary-encoding[2]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3289  moreover AOT_have ‹[λy [F]yx2]↓› by "cqt:2[lambda]"
3290  moreover AOT_have ‹[λy [F]x1y]↓› by "cqt:2[lambda]"
3291  ultimately AOT_have ‹□x1[λy [F]yx2]› and ‹□x2[λy [F]x1y]›
3292    using encoding[axiom_inst, unvarify F] "→E" "&I" by blast+
3293  note A = this
3294  AOT_hence ‹□(x1[λy [F]yx2] & x2[λy [F]x1y])›
3295    using "KBasic:3"[THEN "≡E"(2)] "&I" by blast
3296  AOT_thus ‹□x1x2[F]›
3297    by (rule "nary-encoding[2]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3298qed
3299
3300AOT_theorem "pre-en-eq:1[3]": ‹x1x2x3[F] → □x1x2x3[F]›
3301proof (rule "→I")
3302  AOT_assume ‹x1x2x3[F]›
3303  AOT_hence ‹x1[λy [F]yx2x3]› and ‹x2[λy [F]x1yx3]› and ‹x3[λy [F]x1x2y]›
3304    using "nary-encoding[3]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3305  moreover AOT_have ‹[λy [F]yx2x3]↓› by "cqt:2[lambda]"
3306  moreover AOT_have ‹[λy [F]x1yx3]↓› by "cqt:2[lambda]"
3307  moreover AOT_have ‹[λy [F]x1x2y]↓› by "cqt:2[lambda]"
3308  ultimately AOT_have ‹□x1[λy [F]yx2x3]› and ‹□x2[λy [F]x1yx3]› and ‹□x3[λy [F]x1x2y]›
3309    using encoding[axiom_inst, unvarify F] "→E" by blast+
3310  note A = this
3311  AOT_have B: ‹□(x1[λy [F]yx2x3] & x2[λy [F]x1yx3] & x3[λy [F]x1x2y])›
3312    by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3313  AOT_thus ‹□x1x2x3[F]›
3314    by (rule "nary-encoding[3]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3315qed
3316
3317AOT_theorem "pre-en-eq:1[4]": ‹x1x2x3x4[F] → □x1x2x3x4[F]›
3318proof (rule "→I")
3319  AOT_assume ‹x1x2x3x4[F]›
3320  AOT_hence ‹x1[λy [F]yx2x3x4]› and ‹x2[λy [F]x1yx3x4]› and ‹x3[λy [F]x1x2yx4]› and  ‹x4[λy [F]x1x2x3y]›
3321    using "nary-encoding[4]"[axiom_inst, THEN "≡E"(1)] "&E" by metis+
3322  moreover AOT_have ‹[λy [F]yx2x3x4]↓› by "cqt:2[lambda]"
3323  moreover AOT_have ‹[λy [F]x1yx3x4]↓› by "cqt:2[lambda]"
3324  moreover AOT_have ‹[λy [F]x1x2yx4]↓› by "cqt:2[lambda]"
3325  moreover AOT_have ‹[λy [F]x1x2x3y]↓› by "cqt:2[lambda]"
3326  ultimately AOT_have ‹□x1[λy [F]yx2x3x4]› and ‹□x2[λy [F]x1yx3x4]› and ‹□x3[λy [F]x1x2yx4]› and ‹□x4[λy [F]x1x2x3y]›
3327    using "→E" encoding[axiom_inst, unvarify F] by blast+
3328  note A = this
3329  AOT_have B: ‹□(x1[λy [F]yx2x3x4] & x2[λy [F]x1yx3x4] & x3[λy [F]x1x2yx4] & x4[λy [F]x1x2x3y])›
3330    by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3331  AOT_thus ‹□x1x2x3x4[F]›
3332    by (rule "nary-encoding[4]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3333qed
3334
3335AOT_theorem "pre-en-eq:2[1]": ‹¬x1[F] → □¬x1[F]›
3336proof (rule "→I"; rule "raa-cor:1")
3337  AOT_assume ‹¬□¬x1[F]›
3338  AOT_hence ‹◇x1[F]›
3339    by (rule "conventions:5"[THEN "≡dfI"])
3340  AOT_hence ‹x1[F]›
3341    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[1]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3342  moreover AOT_assume ‹¬x1[F]›
3343  ultimately AOT_show ‹x1[F] & ¬x1[F]› by (rule "&I")
3344qed
3345AOT_theorem "pre-en-eq:2[2]": ‹¬x1x2[F] → □¬x1x2[F]›
3346proof (rule "→I"; rule "raa-cor:1")
3347  AOT_assume ‹¬□¬x1x2[F]›
3348  AOT_hence ‹◇x1x2[F]›
3349    by (rule "conventions:5"[THEN "≡dfI"])
3350  AOT_hence ‹x1x2[F]›
3351    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[2]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3352  moreover AOT_assume ‹¬x1x2[F]›
3353  ultimately AOT_show ‹x1x2[F] & ¬x1x2[F]› by (rule "&I")
3354qed
3355
3356AOT_theorem "pre-en-eq:2[3]": ‹¬x1x2x3[F] → □¬x1x2x3[F]›
3357proof (rule "→I"; rule "raa-cor:1")
3358  AOT_assume ‹¬□¬x1x2x3[F]›
3359  AOT_hence ‹◇x1x2x3[F]›
3360    by (rule "conventions:5"[THEN "≡dfI"])
3361  AOT_hence ‹x1x2x3[F]›
3362    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[3]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3363  moreover AOT_assume ‹¬x1x2x3[F]›
3364  ultimately AOT_show ‹x1x2x3[F] & ¬x1x2x3[F]› by (rule "&I")
3365qed
3366
3367AOT_theorem "pre-en-eq:2[4]": ‹¬x1x2x3x4[F] → □¬x1x2x3x4[F]›
3368proof (rule "→I"; rule "raa-cor:1")
3369  AOT_assume ‹¬□¬x1x2x3x4[F]›
3370  AOT_hence ‹◇x1x2x3x4[F]›
3371    by (rule "conventions:5"[THEN "≡dfI"])
3372  AOT_hence ‹x1x2x3x4[F]›
3373    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[4]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3374  moreover AOT_assume ‹¬x1x2x3x4[F]›
3375  ultimately AOT_show ‹x1x2x3x4[F] & ¬x1x2x3x4[F]› by (rule "&I")
3376qed
3377
3378AOT_theorem "en-eq:1[1]": ‹◇x1[F] ≡ □x1[F]›
3379  using "pre-en-eq:1[1]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3380AOT_theorem "en-eq:1[2]": ‹◇x1x2[F] ≡ □x1x2[F]›
3381  using "pre-en-eq:1[2]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3382AOT_theorem "en-eq:1[3]": ‹◇x1x2x3[F] ≡ □x1x2x3[F]›
3383  using "pre-en-eq:1[3]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3384AOT_theorem "en-eq:1[4]": ‹◇x1x2x3x4[F] ≡ □x1x2x3x4[F]›
3385  using "pre-en-eq:1[4]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3386
3387AOT_theorem "en-eq:2[1]": ‹x1[F] ≡ □x1[F]›
3388  by (simp add: "≡I" "pre-en-eq:1[1]" "qml:2"[axiom_inst])
3389AOT_theorem "en-eq:2[2]": ‹x1x2[F] ≡ □x1x2[F]›
3390  by (simp add: "≡I" "pre-en-eq:1[2]" "qml:2"[axiom_inst])
3391AOT_theorem "en-eq:2[3]": ‹x1x2x3[F] ≡ □x1x2x3[F]›
3392  by (simp add: "≡I" "pre-en-eq:1[3]" "qml:2"[axiom_inst])
3393AOT_theorem "en-eq:2[4]": ‹x1x2x3x4[F] ≡ □x1x2x3x4[F]›
3394  by (simp add: "≡I" "pre-en-eq:1[4]" "qml:2"[axiom_inst])
3395
3396AOT_theorem "en-eq:3[1]": ‹◇x1[F] ≡ x1[F]›
3397  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[1]"] "≡I" by blast
3398AOT_theorem "en-eq:3[2]": ‹◇x1x2[F] ≡ x1x2[F]›
3399  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[2]"] "≡I" by blast
3400AOT_theorem "en-eq:3[3]": ‹◇x1x2x3[F] ≡ x1x2x3[F]›
3401  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[3]"] "≡I" by blast
3402AOT_theorem "en-eq:3[4]": ‹◇x1x2x3x4[F] ≡ x1x2x3x4[F]›
3403  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[4]"] "≡I" by blast
3404
3405AOT_theorem "en-eq:4[1]": ‹(x1[F] ≡ y1[G]) ≡ (□x1[F] ≡ □y1[G])›
3406  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3407  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[1]" by blast+
3408AOT_theorem "en-eq:4[2]": ‹(x1x2[F] ≡ y1y2[G]) ≡ (□x1x2[F] ≡ □y1y2[G])›
3409  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3410  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[2]" by blast+
3411AOT_theorem "en-eq:4[3]": ‹(x1x2x3[F] ≡ y1y2y3[G]) ≡ (□x1x2x3[F] ≡ □y1y2y3[G])›
3412  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3413  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[3]" by blast+
3414AOT_theorem "en-eq:4[4]": ‹(x1x2x3x4[F] ≡ y1y2y3y4[G]) ≡ (□x1x2x3x4[F] ≡ □y1y2y3y4[G])›
3415  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3416  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[4]" by blast+
3417
3418AOT_theorem "en-eq:5[1]": ‹□(x1[F] ≡ y1[G]) ≡ (□x1[F] ≡ □y1[G])›
3419  apply (rule "≡I"; rule "→I")
3420  using "en-eq:4[1]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3421  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3422        "&I"[OF "pre-en-eq:1[1]"[THEN RN], OF "pre-en-eq:1[1]"[THEN RN]] by blast
3423AOT_theorem "en-eq:5[2]": ‹□(x1x2[F] ≡ y1y2[G]) ≡ (□x1x2[F] ≡ □y1y2[G])›
3424  apply (rule "≡I"; rule "→I")
3425  using "en-eq:4[2]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3426  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3427        "&I"[OF "pre-en-eq:1[2]"[THEN RN], OF "pre-en-eq:1[2]"[THEN RN]] by blast
3428AOT_theorem "en-eq:5[3]": ‹□(x1x2x3[F] ≡ y1y2y3[G]) ≡ (□x1x2x3[F] ≡ □y1y2y3[G])›
3429  apply (rule "≡I"; rule "→I")
3430  using "en-eq:4[3]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3431  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3432        "&I"[OF "pre-en-eq:1[3]"[THEN RN], OF "pre-en-eq:1[3]"[THEN RN]] by blast
3433AOT_theorem "en-eq:5[4]": ‹□(x1x2x3x4[F] ≡ y1y2y3y4[G]) ≡ (□x1x2x3x4[F] ≡ □y1y2y3y4[G])›
3434  apply (rule "≡I"; rule "→I")
3435  using "en-eq:4[4]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3436  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3437        "&I"[OF "pre-en-eq:1[4]"[THEN RN], OF "pre-en-eq:1[4]"[THEN RN]] by blast
3438
3439AOT_theorem "en-eq:6[1]": ‹(x1[F] ≡ y1[G]) ≡ □(x1[F] ≡ y1[G])›
3440  using "en-eq:5[1]"[symmetric] "en-eq:4[1]" "≡E"(5) by fast
3441AOT_theorem "en-eq:6[2]": ‹(x1x2[F] ≡ y1y2[G]) ≡ □(x1x2[F] ≡ y1y2[G])›
3442  using "en-eq:5[2]"[symmetric] "en-eq:4[2]" "≡E"(5) by fast
3443AOT_theorem "en-eq:6[3]": ‹(x1x2x3[F] ≡ y1y2y3[G]) ≡ □(x1x2x3[F] ≡ y1y2y3[G])›
3444  using "en-eq:5[3]"[symmetric] "en-eq:4[3]" "≡E"(5) by fast
3445AOT_theorem "en-eq:6[4]": ‹(x1x2x3x4[F] ≡ y1y2y3y4[G]) ≡ □(x1x2x3x4[F] ≡ y1y2y3y4[G])›
3446  using "en-eq:5[4]"[symmetric] "en-eq:4[4]" "≡E"(5) by fast
3447
3448AOT_theorem "en-eq:7[1]": ‹¬x1[F] ≡ □¬x1[F]›
3449  using "pre-en-eq:2[1]" "qml:2"[axiom_inst] "≡I" by blast
3450AOT_theorem "en-eq:7[2]": ‹¬x1x2[F] ≡ □¬x1x2[F]›
3451  using "pre-en-eq:2[2]" "qml:2"[axiom_inst] "≡I" by blast
3452AOT_theorem "en-eq:7[3]": ‹¬x1x2x3[F] ≡ □¬x1x2x3[F]›
3453  using "pre-en-eq:2[3]" "qml:2"[axiom_inst] "≡I" by blast
3454AOT_theorem "en-eq:7[4]": ‹¬x1x2x3x4[F] ≡ □¬x1x2x3x4[F]›
3455  using "pre-en-eq:2[4]" "qml:2"[axiom_inst] "≡I" by blast
3456
3457AOT_theorem "en-eq:8[1]": ‹◇¬x1[F] ≡ ¬x1[F]›
3458  using "en-eq:2[1]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3459AOT_theorem "en-eq:8[2]": ‹◇¬x1x2[F] ≡ ¬x1x2[F]›
3460  using "en-eq:2[2]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3461AOT_theorem "en-eq:8[3]": ‹◇¬x1x2x3[F] ≡ ¬x1x2x3[F]›
3462  using "en-eq:2[3]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3463AOT_theorem "en-eq:8[4]": ‹◇¬x1x2x3x4[F] ≡ ¬x1x2x3x4[F]›
3464  using "en-eq:2[4]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3465
3466AOT_theorem "en-eq:9[1]": ‹◇¬x1[F] ≡ □¬x1[F]›
3467  using "en-eq:7[1]" "en-eq:8[1]" "≡E"(5) by blast
3468AOT_theorem "en-eq:9[2]": ‹◇¬x1x2[F] ≡ □¬x1x2[F]›
3469  using "en-eq:7[2]" "en-eq:8[2]" "≡E"(5) by blast
3470AOT_theorem "en-eq:9[3]": ‹◇¬x1x2x3[F] ≡ □¬x1x2x3[F]›
3471  using "en-eq:7[3]" "en-eq:8[3]" "≡E"(5) by blast
3472AOT_theorem "en-eq:9[4]": ‹◇¬x1x2x3x4[F] ≡ □¬x1x2x3x4[F]›
3473  using "en-eq:7[4]" "en-eq:8[4]" "≡E"(5) by blast
3474
3475AOT_theorem "en-eq:10[1]": ‹𝒜x1[F] ≡ x1[F]›
3476  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[1]" "pre-en-eq:1[1]")
3477AOT_theorem "en-eq:10[2]": ‹𝒜x1x2[F] ≡ x1x2[F]›
3478  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[2]" "pre-en-eq:1[2]")
3479AOT_theorem "en-eq:10[3]": ‹𝒜x1x2x3[F] ≡ x1x2x3[F]›
3480  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[3]" "pre-en-eq:1[3]")
3481AOT_theorem "en-eq:10[4]": ‹𝒜x1x2x3x4[F] ≡ x1x2x3x4[F]›
3482  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[4]" "pre-en-eq:1[4]")
3483
3484AOT_theorem "oa-facts:1": ‹O!x → □O!x›
3485proof(rule "→I")
3486  AOT_modally_strict {
3487    AOT_have ‹[λx ◇E!x]x ≡ ◇E!x›
3488      by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3489  } note θ = this
3490  AOT_assume ‹O!x›
3491  AOT_hence ‹[λx ◇E!x]x›
3492    by (rule "=dfE"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3493  AOT_hence ‹◇E!x› using θ[THEN "≡E"(1)] by blast
3494  AOT_hence ‹□◇E!x› using "qml:3"[axiom_inst, THEN "→E"] by blast
3495  AOT_hence ‹□[λx ◇E!x]x›
3496    by (AOT_subst ‹[λx ◇E!x]x› ‹◇E!x›)
3497       (auto simp: θ)
3498  AOT_thus ‹□O!x›
3499    by (rule "=dfI"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3500qed
3501
3502AOT_theorem "oa-facts:2": ‹A!x → □A!x›
3503proof(rule "→I")
3504  AOT_modally_strict {
3505    AOT_have ‹[λx ¬◇E!x]x ≡ ¬◇E!x›
3506      by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3507  } note θ = this
3508  AOT_assume ‹A!x›
3509  AOT_hence ‹[λx ¬◇E!x]x›
3510    by (rule "=dfE"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3511  AOT_hence ‹¬◇E!x› using θ[THEN "≡E"(1)] by blast
3512  AOT_hence ‹□¬E!x› using "KBasic2:1"[THEN "≡E"(2)] by blast
3513  AOT_hence ‹□□¬E!x› using "4"[THEN "→E"] by blast
3514  AOT_hence ‹□¬◇E!x›
3515    using "KBasic2:1"
3516    by (AOT_subst (reverse) ‹¬◇E!x› ‹□¬E!x›) blast
3517  AOT_hence ‹□[λx ¬◇E!x]x›
3518    by (AOT_subst ‹[λx ¬◇E!x]x› ‹¬◇E!x›)
3519       (auto simp: θ)
3520  AOT_thus ‹□A!x›
3521    by (rule "=dfI"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3522qed
3523
3524AOT_theorem "oa-facts:3": ‹◇O!x → O!x›
3525  using "oa-facts:1" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3526AOT_theorem "oa-facts:4": ‹◇A!x → A!x›
3527  using "oa-facts:2" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3528
3529AOT_theorem "oa-facts:5": ‹◇O!x ≡ □O!x›
3530  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3531
3532AOT_theorem "oa-facts:6": ‹◇A!x ≡ □A!x›
3533  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3534
3535AOT_theorem "oa-facts:7": ‹O!x ≡ 𝒜O!x›
3536  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3537
3538AOT_theorem "oa-facts:8": ‹A!x ≡ 𝒜A!x›
3539  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3540
3541AOT_theorem "beta-C-meta": ‹[λμ1...μn φ{μ1...μn, ν1...νn}]↓ → ([λμ1...μn φ{μ1...μn, ν1...νn}]ν1...νn ≡ φ{ν1...νn, ν1...νn})›
3542  using "lambda-predicates:2"[axiom_inst] by blast
3543
3544AOT_theorem "beta-C-cor:1": ‹(∀ν1...∀νn([λμ1...μn φ{μ1...μn, ν1...νn}]↓)) → ∀ν1...∀νn ([λμ1...μn φ{μ1...μn, ν1...νn}]ν1...νn ≡ φ{ν1...νn, ν1...νn})›
3545  apply (rule "cqt-basic:14"[where 'a='a, THEN "→E"])
3546  using "beta-C-meta" "∀I" by fast
3547
3548AOT_theorem "beta-C-cor:2": ‹[λμ1...μn φ{μ1...μn}]↓ → ∀ν1...∀νn ([λμ1...μn φ{μ1...μn}]ν1...νn ≡ φ{ν1...νn})›
3549  apply (rule "→I"; rule "∀I")
3550  using "beta-C-meta"[THEN "→E"] by fast
3551
3552(* TODO: syntax + double-check if this is really a faithful representation *)
3553theorem "beta-C-cor:3": assumes ‹⋀ν1νn. AOT_instance_of_cqt_2 (φ (AOT_term_of_var ν1νn))›
3554  shows ‹[v ⊨ ∀ν1...∀νn ([λμ1...μn φ{ν1...νn,μ1...μn}]ν1...νn ≡ φ{ν1...νn,ν1...νn})]›
3555  using "cqt:2[lambda]"[axiom_inst, OF assms] "beta-C-cor:1"[THEN "→E"] "∀I" by fast
3556
3557AOT_theorem "betaC:1:a": ‹[λμ1...μn φ{μ1...μn}]κ1...κn ⊢□ φ{κ1...κn}›
3558proof -
3559  AOT_modally_strict {
3560    AOT_assume ‹[λμ1...μn φ{μ1...μn}]κ1...κn›
3561    moreover AOT_have ‹[λμ1...μn φ{μ1...μn}]↓› and ‹κ1...κn↓›
3562      using calculation "cqt:5:a"[axiom_inst, THEN "→E"] "&E" by blast+
3563    ultimately AOT_show ‹φ{κ1...κn}›
3564      using "beta-C-cor:2"[THEN "→E", THEN "∀E"(1), THEN "≡E"(1)] by blast
3565  }
3566qed
3567
3568AOT_theorem "betaC:1:b": ‹¬φ{κ1...κn} ⊢□ ¬[λμ1...μn φ{μ1...μn}]κ1...κn›
3569  using "betaC:1:a" "raa-cor:3" by blast
3570
3571lemmas "β→C" = "betaC:1:a" "betaC:1:b"
3572
3573AOT_theorem "betaC:2:a": ‹[λμ1...μn φ{μ1...μn}]↓, κ1...κn↓, φ{κ1...κn} ⊢□ [λμ1...μn φ{μ1...μn}]κ1...κn›
3574proof -
3575  AOT_modally_strict {
3576    AOT_assume 1: ‹[λμ1...μn φ{μ1...μn}]↓› and 2: ‹κ1...κn↓› and 3: ‹φ{κ1...κn}›
3577    AOT_hence ‹[λμ1...μn φ{μ1...μn}]κ1...κn›
3578      using "beta-C-cor:2"[THEN "→E", OF 1, THEN "∀E"(1), THEN "≡E"(2)] by blast
3579  }
3580  AOT_thus ‹[λμ1...μn φ{μ1...μn}]↓, κ1...κn↓, φ{κ1...κn} ⊢□ [λμ1...μn φ{μ1...μn}]κ1...κn›
3581    by blast
3582qed
3583
3584AOT_theorem "betaC:2:b": ‹[λμ1...μn φ{μ1...μn}]↓, κ1...κn↓, ¬[λμ1...μn φ{μ1...μn}]κ1...κn ⊢□ ¬φ{κ1...κn}›
3585  using "betaC:2:a" "raa-cor:3" by blast
3586
3587lemmas "β←C" = "betaC:2:a" "betaC:2:b"
3588
3589AOT_theorem "eta-conversion-lemma1:1": ‹Π↓ → [λx1...xn [Π]x1...xn] = Π›
3590  using "lambda-predicates:3"[axiom_inst] "∀I" "∀E"(1) "→I" by fast
3591
3592AOT_theorem "eta-conversion-lemma1:2": ‹Π↓ → [λν1...νn [Π]ν1...νn] = Π›
3593  using "eta-conversion-lemma1:1". (* TODO: spurious in the embedding *)
3594
3595(* match (τ) in "λa . ?b" ⇒ ‹match (τ') in "λa . ?b" ⇒ ‹fail›› ¦ _ ⇒ ‹ *)
3596
3597text‹Note: not explicitly part of PLM.›
3598AOT_theorem id_sym: assumes ‹τ = τ'› shows ‹τ' = τ›
3599  using "rule=E"[where φ="λ τ' . «τ' = τ»", rotated 1, OF assms]
3600        "=I"(1)[OF "t=t-proper:1"[THEN "→E", OF assms]] by auto
3601declare id_sym[sym]
3602
3603text‹Note: not explicitly part of PLM.›
3604AOT_theorem id_trans: assumes ‹τ = τ'› and ‹τ' = τ''› shows ‹τ = τ''›
3605  using "rule=E" assms by blast
3606declare id_trans[trans]
3607
3608method "ηC" for Π :: ‹<'a::{AOT_Term_id_2,AOT_κs}>› = (match conclusion in "[v ⊨ τ{Π} = τ'{Π}]" for v τ τ' ⇒ ‹
3609rule "rule=E"[rotated 1, OF "eta-conversion-lemma1:2"[THEN "→E", of v "«[Π]»", symmetric]]
3610›)
3611(*
3612AOT_theorem ‹[λy [λz [P]z]y → [λu [S]u]y] = [λy [P]y → [S]y]›
3613  apply ("ηC" "«[P]»") defer
3614   apply ("ηC" "«[S]»") defer
3615  oops
3616*)
3617(* TODO: proper representation of eta_conversion_lemma2 *)
3618
3619AOT_theorem "sub-des-lam:1": ‹[λz1...zn  χ{z1...zn, ιx φ{x}}]↓ & ιx φ{x} = ιx ψ{x} → [λz1...zn χ{z1...zn, ιx φ{x}}] = [λz1...zn χ{z1...zn, ιx ψ{x}}]›
3620proof(rule "→I")
3621  AOT_assume A: ‹[λz1...zn  χ{z1...zn, ιx φ{x}}]↓ & ιx φ{x} = ιx ψ{x}›
3622  AOT_show ‹[λz1...zn χ{z1...zn, ιx φ{x}}] = [λz1...zn χ{z1...zn, ιx ψ{x}}]›
3623    using "rule=E"[where φ="λ τ . «[λz1...zn χ{z1...zn, ιx φ{x}}] = [λz1...zn χ{z1...zn, τ}]»",
3624               OF "=I"(1)[OF A[THEN "&E"(1)]], OF A[THEN "&E"(2)]]
3625    by blast
3626qed
3627
3628AOT_theorem "sub-des-lam:2": ‹ιx φ{x} = ιx ψ{x} → χ{ιx φ{x}} = χ{ιx ψ{x}}› for χ :: ‹κ ⇒ 𝗈›
3629  using "rule=E"[where φ="λ τ . «χ{ιx φ{x}} = χ{τ}»", OF "=I"(1)[OF "log-prop-prop:2"]] "→I" by blast
3630
3631AOT_theorem "prop-equiv": ‹F = G ≡ ∀x (x[F] ≡ x[G])›
3632proof(rule "≡I"; rule "→I")
3633  AOT_assume ‹F = G›
3634  AOT_thus ‹∀x (x[F] ≡ x[G])›
3635    by (rule "rule=E"[rotated]) (fact "oth-class-taut:3:a"[THEN GEN])
3636next
3637  AOT_assume ‹∀x (x[F] ≡ x[G])›
3638  AOT_hence ‹x[F] ≡ x[G]› for x using "∀E" by blast
3639  AOT_hence ‹□(x[F] ≡ x[G])› for x using "en-eq:6[1]"[THEN "≡E"(1)] by blast
3640  AOT_hence ‹∀x □(x[F] ≡ x[G])› by (rule GEN)
3641  AOT_hence ‹□∀x (x[F] ≡ x[G])› using BF[THEN "→E"] by fast
3642  AOT_thus "F = G" using "p-identity-thm2:1"[THEN "≡E"(2)] by blast
3643qed
3644
3645AOT_theorem "relations:1":
3646  assumes ‹INSTANCE_OF_CQT_2(φ)›
3647  shows ‹∃F □∀x1...∀xn ([F]x1...xn ≡ φ{x1...xn})›
3648  apply (rule "∃I"(1)[where τ="«[λx1...xn φ{x1...xn}]»"])
3649  using "cqt:2[lambda]"[OF assms, axiom_inst] "beta-C-cor:2"[THEN "→E", THEN RN] by blast+
3650
3651AOT_theorem "relations:2":
3652  assumes ‹INSTANCE_OF_CQT_2(φ)›
3653  shows ‹∃F □∀x ([F]x ≡ φ{x})›
3654  using "relations:1" assms by blast
3655
3656AOT_theorem "block-paradox:1": ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
3657proof(rule RAA(2))
3658  let ?φ="λ τ. «∃G (τ[G] & ¬[G]τ)»"
3659  AOT_assume A: ‹[λx «?φ x»]↓›
3660  AOT_have ‹∃x (A!x & ∀F (x[F] ≡ F = [λx «?φ x»]))›
3661    using "A-objects"[axiom_inst] by fast
3662  then AOT_obtain a where ξ: ‹A!a & ∀F (a[F] ≡ F = [λx «?φ x»])›
3663    using "∃E"[rotated] by blast
3664  AOT_show ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
3665  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3666    AOT_assume B: ‹[λx «?φ x»]a›
3667    AOT_hence ‹∃G (a[G] & ¬[G]a)› using "β→C" A by blast
3668    then AOT_obtain P where ‹a[P] & ¬[P]a› using "∃E"[rotated] by blast
3669    moreover AOT_have ‹P = [λx «?φ x»]›
3670      using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3671    ultimately AOT_have ‹¬[λx «?φ x»]a›
3672      using "rule=E" "&E"(2) by fast
3673    AOT_thus ‹¬[λx ∃G (x[G] & ¬[G]x)]↓› using B RAA by blast
3674  next
3675    AOT_assume B: ‹¬[λx «?φ x»]a›
3676    AOT_hence ‹¬∃G (a[G] & ¬[G]a)› using "β←C" "cqt:2[const_var]"[of a, axiom_inst] A by blast
3677    AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)› using "cqt-further:4"[THEN "→E"] by blast
3678    AOT_have ‹∀G (a[G] → [G]a)›
3679      by (AOT_subst ‹a[G] → [G]a› ‹¬(a[G] & ¬[G]a)› for: G)
3680         (auto simp: "oth-class-taut:1:a" C)
3681    AOT_hence ‹a[λx «?φ x»] → [λx «?φ x»]a› using "∀E" A by blast
3682    moreover AOT_have ‹a[λx «?φ x»]› using ξ[THEN "&E"(2), THEN "∀E"(1), OF A, THEN "≡E"(2)]
3683      using "=I"(1)[OF A] by blast
3684    ultimately AOT_show ‹¬[λx ∃G (x[G] & ¬[G]x)]↓› using B "→E" RAA by blast
3685  qed
3686qed(simp)
3687
3688AOT_theorem "block-paradox:2": ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
3689proof(rule RAA(2))
3690  AOT_assume ‹∃F ∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))›
3691  then AOT_obtain F where F_prop: ‹∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))› using "∃E"[rotated] by blast
3692  AOT_have ‹∃x (A!x & ∀G (x[G] ≡ G = F))›
3693    using "A-objects"[axiom_inst] by fast
3694  then AOT_obtain a where ξ: ‹A!a & ∀G (a[G] ≡ G = F)›
3695    using "∃E"[rotated] by blast
3696  AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
3697  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3698    AOT_assume B: ‹[F]a›
3699    AOT_hence ‹∃G (a[G] & ¬[G]a)› using F_prop[THEN "∀E"(2), THEN "≡E"(1)] by blast
3700    then AOT_obtain P where ‹a[P] & ¬[P]a› using "∃E"[rotated] by blast
3701    moreover AOT_have ‹P = F›
3702      using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3703    ultimately AOT_have ‹¬[F]a›
3704      using "rule=E" "&E"(2) by fast
3705    AOT_thus ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))› using B RAA by blast
3706  next
3707    AOT_assume B: ‹¬[F]a›
3708    AOT_hence ‹¬∃G (a[G] & ¬[G]a)›
3709      using "oth-class-taut:4:b"[THEN "≡E"(1), OF F_prop[THEN "∀E"(2)[of _ _ a]], THEN "≡E"(1)] by simp
3710    AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)› using "cqt-further:4"[THEN "→E"] by blast
3711    AOT_have ‹∀G (a[G] → [G]a)›
3712      by (AOT_subst ‹a[G] → [G]a› ‹¬(a[G] & ¬[G]a)› for: G)
3713         (auto simp: "oth-class-taut:1:a" C)
3714    AOT_hence ‹a[F] → [F]a› using "∀E" by blast
3715    moreover AOT_have ‹a[F]› using ξ[THEN "&E"(2), THEN "∀E"(2), of F, THEN "≡E"(2)]
3716      using "=I"(2) by blast
3717    ultimately AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))› using B "→E" RAA by blast
3718  qed
3719qed(simp)
3720
3721AOT_theorem "block-paradox:3": ‹¬∀y [λz z = y]↓›
3722proof(rule RAA(2))
3723  AOT_assume θ: ‹∀y [λz z = y]↓›
3724  AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃y(F = [λz z = y] & ¬y[F])))›
3725    using "A-objects"[axiom_inst] by force
3726  then AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y (F = [λz z = y] & ¬y[F]))›
3727    using "∃E"[rotated] by blast
3728  AOT_have ζ: ‹a[λz z = a] ≡ ∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3729    using θ[THEN "∀E"(2)] a_prop[THEN "&E"(2), THEN "∀E"(1)] by blast
3730  AOT_show ‹¬∀y [λz z = y]↓›
3731  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3732    AOT_assume A: ‹a[λz z = a]›
3733    AOT_hence ‹∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3734      using ζ[THEN "≡E"(1)] by blast
3735    then AOT_obtain b where b_prop: ‹[λz z = a] = [λz z = b] & ¬b[λz z = a]›
3736      using "∃E"[rotated] by blast
3737    moreover AOT_have ‹a = a› by (rule "=I")
3738    moreover AOT_have ‹[λz z = a]↓› using θ "∀E" by blast
3739    moreover AOT_have ‹a↓› using "cqt:2[const_var]"[axiom_inst] .
3740    ultimately AOT_have ‹[λz z = a]a› using "β←C" by blast
3741    AOT_hence ‹[λz z = b]a› using "rule=E" b_prop[THEN "&E"(1)] by fast
3742    AOT_hence ‹a = b› using "β→C" by blast
3743    AOT_hence ‹b[λz z = a]› using A "rule=E" by fast
3744    AOT_thus ‹¬∀y [λz z = y]↓› using b_prop[THEN "&E"(2)] RAA by blast
3745  next
3746    AOT_assume A: ‹¬a[λz z = a]›
3747    AOT_hence ‹¬∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3748      using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3749    AOT_hence ‹∀y ¬([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3750      using "cqt-further:4"[THEN "→E"] by blast
3751    AOT_hence ‹¬([λz z = a] = [λz z = a] & ¬a[λz z = a])›
3752      using "∀E" by blast
3753    AOT_hence ‹[λz z = a] = [λz z = a] → a[λz z = a]›
3754      by (metis "&I" "deduction-theorem" "raa-cor:4")
3755    AOT_hence ‹a[λz z = a]› using "=I"(1) θ[THEN "∀E"(2)] "→E" by blast
3756    AOT_thus ‹¬∀y [λz z = y]↓› using A RAA by blast
3757  qed
3758qed(simp)
3759
3760AOT_theorem "block-paradox:4": ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
3761proof(rule RAA(2))
3762  AOT_assume θ: ‹∀y ∃F ∀x([F]x ≡ x = y)›
3763  AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃z (∀y([F]y ≡ y = z) & ¬z[F])))›
3764    using "A-objects"[axiom_inst] by force
3765  then AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃z (∀y([F]y ≡ y = z) & ¬z[F]))›
3766    using "∃E"[rotated] by blast
3767  AOT_obtain F where F_prop: ‹∀x ([F]x ≡ x = a)› using θ[THEN "∀E"(2)] "∃E"[rotated] by blast
3768  AOT_have ζ: ‹a[F] ≡ ∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3769    using a_prop[THEN "&E"(2), THEN "∀E"(2)] by blast
3770  AOT_show ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
3771  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3772    AOT_assume A: ‹a[F]›
3773    AOT_hence ‹∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3774      using ζ[THEN "≡E"(1)] by blast
3775    then AOT_obtain b where b_prop: ‹∀y ([F]y ≡ y = b) & ¬b[F]›
3776      using "∃E"[rotated] by blast
3777    moreover AOT_have ‹[F]a› using F_prop[THEN "∀E"(2), THEN "≡E"(2)] "=I"(2) by blast
3778    ultimately AOT_have ‹a = b› using "∀E"(2) "≡E"(1) "&E" by fast
3779    AOT_hence ‹a = b› using "β→C" by blast
3780    AOT_hence ‹b[F]› using A "rule=E" by fast
3781    AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)› using b_prop[THEN "&E"(2)] RAA by blast
3782  next
3783    AOT_assume A: ‹¬a[F]›
3784    AOT_hence ‹¬∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3785      using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3786    AOT_hence ‹∀z ¬(∀y ([F]y ≡ y = z) & ¬z[F])›
3787      using "cqt-further:4"[THEN "→E"] by blast
3788    AOT_hence ‹¬(∀y ([F]y ≡ y = a) & ¬a[F])›
3789      using "∀E" by blast
3790    AOT_hence ‹∀y ([F]y ≡ y = a) → a[F]›
3791      by (metis "&I" "deduction-theorem" "raa-cor:4")
3792    AOT_hence ‹a[F]› using F_prop "→E" by blast
3793    AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)› using A RAA by blast
3794  qed
3795qed(simp)
3796
3797AOT_theorem "block-paradox:5": ‹¬∃F∀x∀y([F]xy ≡ y = x)›
3798proof(rule "raa-cor:2")
3799  AOT_assume ‹∃F∀x∀y([F]xy ≡ y = x)›
3800  then AOT_obtain F where F_prop: ‹∀x∀y([F]xy ≡ y = x)› using "∃E"[rotated] by blast
3801  {
3802    fix x
3803    AOT_have 1: ‹∀y([F]xy ≡ y = x)› using F_prop "∀E" by blast
3804    AOT_have 2: ‹[λz [F]xz]↓› by "cqt:2[lambda]"
3805    moreover AOT_have ‹∀y([λz [F]xz]y ≡ y = x)›
3806    proof(rule "∀I")
3807      fix y
3808      AOT_have ‹[λz [F]xz]y ≡ [F]xy›
3809        using "beta-C-meta"[THEN "→E"] 2 by fast
3810      also AOT_have ‹... ≡ y = x› using 1 "∀E"
3811        by fast
3812      finally AOT_show ‹[λz [F]xz]y ≡ y = x›.
3813    qed
3814    ultimately AOT_have ‹∃F∀y([F]y ≡ y = x)›
3815      using "∃I" by fast
3816  }
3817  AOT_hence ‹∀x∃F∀y([F]y ≡ y = x)›
3818    by (rule GEN)
3819  AOT_thus ‹∀x∃F∀y([F]y ≡ y = x) & ¬∀x∃F∀y([F]y ≡ y = x)›
3820    using "&I" "block-paradox:4" by blast
3821qed
3822
3823AOT_act_theorem "block-paradox2:1": ‹∀x [G]x → ¬[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3824proof(rule "→I"; rule "raa-cor:2")
3825  AOT_assume antecedant: ‹∀x [G]x›
3826  AOT_have Lemma: ‹∀x ([G]ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))›
3827  proof(rule GEN)
3828    fix x
3829    AOT_have A: ‹[G]ιy (y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3830    proof(rule "≡I"; rule "→I")
3831      AOT_assume ‹[G]ιy (y = x & ∃H (x[H] & ¬[H]x))›
3832      AOT_hence ‹ιy (y = x & ∃H (x[H] & ¬[H]x))↓›
3833        using "cqt:5:a"[axiom_inst, THEN "→E", THEN "&E"(2)] by blast
3834      AOT_thus ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3835        using "1-exists:1"[THEN "≡E"(1)] by blast
3836    next
3837      AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3838      AOT_obtain a where a_1: ‹a = x & ∃H (x[H] & ¬[H]x)› and a_2: ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = a)›
3839        using "uniqueness:1"[THEN "≡dfE", OF A] "&E" "∃E"[rotated] by blast
3840      AOT_have a_3: ‹[G]a›
3841        using antecedant "∀E" by blast
3842      AOT_show ‹[G]ιy (y = x & ∃H (x[H] & ¬[H]x))›
3843        apply (rule "russell-axiom[exe,1].russell-axiom"[THEN "≡E"(2)])
3844        apply (rule "∃I"(2))
3845        using a_1 a_2 a_3 "&I" by blast
3846    qed
3847    also AOT_have B: ‹... ≡ ∃H (x[H] & ¬[H]x)›
3848    proof (rule "≡I"; rule "→I")
3849      AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3850      AOT_obtain a where ‹a = x & ∃H (x[H] & ¬[H]x)›
3851        using "uniqueness:1"[THEN "≡dfE", OF A] "&E" "∃E"[rotated] by blast
3852      AOT_thus ‹∃H (x[H] & ¬[H]x)› using "&E" by blast
3853    next
3854      AOT_assume ‹∃H (x[H] & ¬[H]x)›
3855      AOT_hence ‹x = x & ∃H (x[H] & ¬[H]x)›
3856        using "id-eq:1" "&I" by blast
3857      moreover AOT_have ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = x)›
3858        by (simp add: "Conjunction Simplification"(1) "universal-cor")
3859      ultimately AOT_show ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3860        using "uniqueness:1"[THEN "≡dfI"] "&I" "∃I"(2) by fast
3861    qed
3862    finally AOT_show ‹([G]ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))› .
3863  qed
3864
3865  AOT_assume A: ‹[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3866  AOT_have θ: ‹∀x ([λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]x ≡ [G]ιy(y = x & ∃H (x[H] & ¬[H]x)))›
3867    using "beta-C-meta"[THEN "→E", OF A] "∀I" by fast
3868  AOT_have ‹∀x ([λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]x ≡ ∃H (x[H] & ¬[H]x))›
3869    using θ Lemma "cqt-basic:10"[THEN "→E"] "&I" by fast
3870  AOT_hence ‹∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))›
3871    using "∃I"(1) A by fast
3872  AOT_thus ‹(∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))) & (¬∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x)))›
3873    using "block-paradox:2" "&I" by blast
3874qed
3875
3876AOT_act_theorem "block-paradox2:2": ‹∃G ¬[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3877proof(rule "∃I"(1))
3878  AOT_have 0: ‹[λx ∀p (p →p)]↓›
3879    by "cqt:2[lambda]"
3880  moreover AOT_have ‹∀x [λx ∀p (p →p)]x›
3881    apply (rule GEN)
3882    apply (rule "beta-C-cor:2"[THEN "→E", OF 0, THEN "∀E"(2), THEN "≡E"(2)])
3883    using "if-p-then-p" GEN by fast
3884  moreover AOT_have ‹∀G (∀x [G]x → ¬[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓)›
3885      using "block-paradox2:1" "∀I" by fast
3886  ultimately AOT_show ‹¬[λx [λx ∀p (p →p)]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3887    using "∀E"(1) "→E" by blast
3888qed("cqt:2[lambda]")
3889
3890AOT_theorem propositions: ‹∃p □(p ≡ φ)›
3891proof(rule "∃I"(1))
3892  AOT_show ‹□(φ ≡ φ)›
3893    by (simp add: RN "oth-class-taut:3:a")
3894next
3895  AOT_show ‹φ↓›
3896    by (simp add: "log-prop-prop:2")
3897qed
3898
3899AOT_theorem "pos-not-equiv-ne:1": ‹(◇¬∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)) → F ≠ G›
3900proof (rule "→I")
3901  AOT_assume ‹◇¬∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)›
3902  AOT_hence ‹¬□∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)›
3903    using "KBasic:11"[THEN "≡E"(2)] by blast
3904  AOT_hence ‹¬(F = G)›
3905    using "id-rel-nec-equiv:1" "modus-tollens:1" by blast
3906  AOT_thus ‹F ≠ G›
3907    using "=-infix"[THEN "≡dfI"] by blast
3908qed
3909
3910AOT_theorem "pos-not-equiv-ne:2": ‹(◇¬(φ{F} ≡ φ{G})) → F ≠ G›
3911proof (rule "→I")
3912  AOT_modally_strict {
3913    AOT_have ‹¬(φ{F} ≡ φ{G}) → ¬(F = G)›
3914    proof (rule "→I"; rule "raa-cor:2")
3915      AOT_assume 1: ‹F = G›
3916      AOT_hence ‹φ{F} → φ{G}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3917      moreover {
3918        AOT_have ‹G = F› using 1 id_sym by blast
3919        AOT_hence ‹φ{G} → φ{F}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3920      }
3921      ultimately AOT_have ‹φ{F} ≡ φ{G}› using "≡I" by blast
3922      moreover AOT_assume ‹¬(φ{F} ≡ φ{G})›
3923      ultimately AOT_show ‹(φ{F} ≡ φ{G}) & ¬(φ{F} ≡ φ{G})›
3924        using "&I" by blast
3925    qed
3926  }
3927  AOT_hence ‹◇¬(φ{F} ≡ φ{G}) → ◇¬(F = G)›
3928    using "RM:2[prem]" by blast
3929  moreover AOT_assume ‹◇¬(φ{F} ≡ φ{G})›
3930  ultimately AOT_have 0: ‹◇¬(F = G)› using "→E" by blast
3931  AOT_have ‹◇(F ≠ G)›
3932    by (AOT_subst ‹F ≠ G› ‹¬(F = G)›)
3933       (auto simp: "=-infix" "≡Df" 0)
3934  AOT_thus ‹F ≠ G›
3935    using "id-nec2:3"[THEN "→E"] by blast
3936qed
3937
3938AOT_theorem "pos-not-equiv-ne:2[zero]": ‹(◇¬(φ{p} ≡ φ{q})) → p ≠ q›
3939proof (rule "→I")
3940  AOT_modally_strict {
3941    AOT_have ‹¬(φ{p} ≡ φ{q}) → ¬(p = q)›
3942    proof (rule "→I"; rule "raa-cor:2")
3943      AOT_assume 1: ‹p = q›
3944      AOT_hence ‹φ{p} → φ{q}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3945      moreover {
3946        AOT_have ‹q = p› using 1 id_sym by blast
3947        AOT_hence ‹φ{q} → φ{p}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3948      }
3949      ultimately AOT_have ‹φ{p} ≡ φ{q}› using "≡I" by blast
3950      moreover AOT_assume ‹¬(φ{p} ≡ φ{q})›
3951      ultimately AOT_show ‹(φ{p} ≡ φ{q}) & ¬(φ{p} ≡ φ{q})›
3952        using "&I" by blast
3953    qed
3954  }
3955  AOT_hence ‹◇¬(φ{p} ≡ φ{q}) → ◇¬(p = q)›
3956    using "RM:2[prem]" by blast
3957  moreover AOT_assume ‹◇¬(φ{p} ≡ φ{q})›
3958  ultimately AOT_have 0: ‹◇¬(p = q)› using "→E" by blast
3959  AOT_have ‹◇(p ≠ q)›
3960    by (AOT_subst ‹p ≠ q› ‹¬(p = q)›)
3961       (auto simp: 0 "=-infix" "≡Df")
3962  AOT_thus ‹p ≠ q›
3963    using "id-nec2:3"[THEN "→E"] by blast
3964qed
3965
3966AOT_theorem "pos-not-equiv-ne:3": ‹(¬∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)) → F ≠ G›
3967  using "→I" "pos-not-equiv-ne:1"[THEN "→E"] "T◇"[THEN "→E"] by blast
3968
3969AOT_theorem "pos-not-equiv-ne:4": ‹(¬(φ{F} ≡ φ{G})) → F ≠ G›
3970  using "→I" "pos-not-equiv-ne:2"[THEN "→E"] "T◇"[THEN "→E"] by blast
3971
3972AOT_theorem "pos-not-equiv-ne:4[zero]": ‹(¬(φ{p} ≡ φ{q})) → p ≠ q›
3973  using "→I" "pos-not-equiv-ne:2[zero]"[THEN "→E"] "T◇"[THEN "→E"] by blast
3974
3975AOT_define relation_negation :: "Π ⇒ Π" ("_-")
3976  "df-relation-negation": "[F]- =df [λx1...xn ¬[F]x1...xn]"
3977
3978nonterminal φneg
3979syntax "" :: "φneg ⇒ τ" ("_")
3980syntax "" :: "φneg ⇒ φ" ("'(_')")
3981
3982AOT_define relation_negation_0 :: ‹φ ⇒ φneg› ("'(_')-")
3983  "df-relation-negation[zero]": "(p)- =df [λ ¬p]"
3984
3985AOT_theorem "rel-neg-T:1": ‹[λx1...xn ¬[Π]x1...xn]↓›
3986  by "cqt:2[lambda]"
3987
3988AOT_theorem "rel-neg-T:1[zero]": ‹[λ ¬φ]↓›
3989  using "cqt:2[lambda0]"[axiom_inst] by blast
3990
3991AOT_theorem "rel-neg-T:2": ‹[Π]- = [λx1...xn ¬[Π]x1...xn]›
3992  using "=I"(1)[OF "rel-neg-T:1"]
3993  by (rule "=dfI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"])
3994
3995AOT_theorem "rel-neg-T:2[zero]": ‹(φ)- = [λ ¬φ]›
3996  using "=I"(1)[OF "rel-neg-T:1[zero]"]
3997  by (rule "=dfI"(1)[OF "df-relation-negation[zero]", OF "rel-neg-T:1[zero]"])
3998
3999AOT_theorem "rel-neg-T:3": ‹[Π]-↓›
4000  using "=dfI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"] "rel-neg-T:1" by blast
4001
4002AOT_theorem "rel-neg-T:3[zero]": ‹(φ)-↓›
4003  using "log-prop-prop:2" by blast
4004(*  using "=dfI"(1)[OF "df-relation-negation[zero]", OF "rel-neg-T:1[zero]"] "rel-neg-T:1[zero]" by blast *)
4005
4006(* Note: PLM states the zero place case twice *)
4007AOT_theorem "thm-relation-negation:1": ‹[F]-x1...xn ≡ ¬[F]x1...xn›
4008proof -
4009  AOT_have ‹[F]-x1...xn ≡ [λx1...xn ¬[F]x1...xn]x1...xn›
4010    using "rule=E"[rotated, OF "rel-neg-T:2"] "rule=E"[rotated, OF "rel-neg-T:2"[THEN id_sym]]
4011    "→I" "≡I" by fast
4012  also AOT_have ‹... ≡ ¬[F]x1...xn›
4013    using "beta-C-meta"[THEN "→E", OF "rel-neg-T:1"] by fast
4014  finally show ?thesis.
4015qed
4016
4017AOT_theorem "thm-relation-negation:2": ‹¬[F]-x1...xn ≡ [F]x1...xn›
4018  apply (AOT_subst ‹[F]x1...xn› ‹¬¬[F]x1...xn›)
4019   apply (simp add: "oth-class-taut:3:b")
4020  apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4021  using "thm-relation-negation:1".
4022
4023AOT_theorem "thm-relation-negation:3": ‹((p)-) ≡ ¬p›
4024proof -
4025  AOT_have ‹(p)- = [λ ¬p]› using "rel-neg-T:2[zero]" by blast
4026  AOT_hence ‹((p)-) ≡ [λ ¬p]›
4027    using "df-relation-negation[zero]" "log-prop-prop:2" "oth-class-taut:3:a" "rule-id-df:2:a" by blast
4028  also AOT_have ‹[λ ¬p] ≡ ¬p›
4029    by (simp add: "propositions-lemma:2")
4030  finally show ?thesis.
4031qed
4032
4033AOT_theorem "thm-relation-negation:4": ‹(¬((p)-)) ≡ p›
4034  using "thm-relation-negation:3"[THEN "≡E"(1)]
4035        "thm-relation-negation:3"[THEN "≡E"(2)]
4036        "≡I" "→I" RAA by metis
4037
4038AOT_theorem "thm-relation-negation:5": ‹[F] ≠ [F]-›
4039proof -
4040  AOT_have ‹¬([F] = [F]-)›
4041  proof (rule RAA(2))
4042    AOT_show ‹[F]x1...xn → [F]x1...xn› for x1xn
4043      using "if-p-then-p".
4044  next
4045    AOT_assume ‹[F] = [F]-›
4046    AOT_hence ‹[F]- = [F]› using id_sym by blast
4047    AOT_hence ‹[F]x1...xn ≡ ¬[F]x1...xn› for x1xn
4048      using "rule=E" "thm-relation-negation:1" by fast
4049    AOT_thus ‹¬([F]x1...xn → [F]x1...xn)› for x1xn
4050      using "≡E" RAA by metis
4051  qed
4052  thus ?thesis
4053    using "≡dfI" "=-infix" by blast
4054qed
4055
4056AOT_theorem "thm-relation-negation:6": ‹p ≠ (p)-›
4057proof -
4058  AOT_have ‹¬(p = (p)-)›
4059  proof (rule RAA(2))
4060    AOT_show ‹p → p›
4061      using "if-p-then-p".
4062  next
4063    AOT_assume ‹p = (p)-›
4064    AOT_hence ‹(p)- = p› using id_sym by blast
4065    AOT_hence ‹p ≡ ¬p›
4066      using "rule=E" "thm-relation-negation:3" by fast
4067    AOT_thus ‹¬(p → p)›
4068      using "≡E" RAA by metis
4069  qed
4070  thus ?thesis
4071    using "≡dfI" "=-infix" by blast
4072qed
4073
4074AOT_theorem "thm-relation-negation:7": ‹(p)- = (¬p)›
4075  apply (rule "df-relation-negation[zero]"[THEN "=dfE"(1)])
4076  using "cqt:2[lambda0]"[axiom_inst] "rel-neg-T:2[zero]" "propositions-lemma:1" id_trans by blast+
4077
4078AOT_theorem "thm-relation-negation:8": ‹p = q → (¬p) = (¬q)›
4079proof(rule "→I")
4080  AOT_assume ‹p = q›
4081  moreover AOT_have ‹(¬p)↓› using "log-prop-prop:2".
4082  moreover AOT_have ‹(¬p) = (¬p)› using calculation(2) "=I" by blast
4083  ultimately AOT_show ‹(¬p) = (¬q)›
4084    using "rule=E" by fast
4085qed
4086
4087AOT_theorem "thm-relation-negation:9": ‹p = q → (p)- = (q)-›
4088proof(rule "→I")
4089  AOT_assume ‹p = q›
4090  AOT_hence ‹(¬p) = (¬q)› using "thm-relation-negation:8" "→E" by blast
4091  AOT_thus ‹(p)- = (q)-›
4092    using "thm-relation-negation:7" id_sym id_trans by metis
4093qed
4094
4095AOT_define Necessary :: ‹Π ⇒ φ› ("Necessary'(_')")
4096  "contingent-properties:1": ‹Necessary([F]) ≡df □∀x1...∀xn [F]x1...xn›
4097
4098AOT_define Necessary0 :: ‹φ ⇒ φ› ("Necessary0'(_')")
4099  "contingent-properties:1[zero]": ‹Necessary0(p) ≡df □p›
4100
4101AOT_define Impossible :: ‹Π ⇒ φ› ("Impossible'(_')")
4102  "contingent-properties:2": ‹Impossible([F]) ≡df F↓ & □∀x1...∀xn ¬[F]x1...xn›
4103
4104AOT_define Impossible0 :: ‹φ ⇒ φ› ("Impossible0'(_')")
4105  "contingent-properties:2[zero]": ‹Impossible0(p) ≡df □¬p›
4106
4107AOT_define NonContingent :: ‹Π ⇒ φ› ("NonContingent'(_')")
4108  "contingent-properties:3": ‹NonContingent([F]) ≡df Necessary([F]) ∨ Impossible([F])›
4109
4110AOT_define NonContingent0 :: ‹φ ⇒ φ› ("NonContingent0'(_')")
4111  "contingent-properties:3[zero]": ‹NonContingent0(p) ≡df Necessary0(p) ∨ Impossible0(p)›
4112
4113AOT_define Contingent :: ‹Π ⇒ φ› ("Contingent'(_')")
4114  "contingent-properties:4": ‹Contingent([F]) ≡df F↓ & ¬(Necessary([F]) ∨ Impossible([F]))›
4115
4116AOT_define Contingent0 :: ‹φ ⇒ φ› ("Contingent0'(_')")
4117  "contingent-properties:4[zero]": ‹Contingent0(p) ≡df ¬(Necessary0(p) ∨ Impossible0(p))›
4118
4119
4120AOT_theorem "thm-cont-prop:1": ‹NonContingent([F]) ≡ NonContingent([F]-)›
4121proof (rule "≡I"; rule "→I")
4122  AOT_assume ‹NonContingent([F])›
4123  AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
4124    using "≡dfE"[OF "contingent-properties:3"] by blast
4125  moreover {
4126    AOT_assume ‹Necessary([F])›
4127    AOT_hence ‹□(∀x1...∀xn [F]x1...xn)›
4128      using "≡dfE"[OF "contingent-properties:1"] by blast
4129    moreover AOT_modally_strict {
4130      AOT_assume ‹∀x1...∀xn [F]x1...xn›
4131      AOT_hence ‹[F]x1...xn› for x1xn using "∀E" by blast
4132      AOT_hence ‹¬[F]-x1...xn› for x1xn
4133        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:2" "≡E"(1))
4134      AOT_hence ‹∀x1...∀xn ¬[F]-x1...xn› using "∀I" by fast
4135    }
4136    ultimately AOT_have ‹□(∀x1...∀xn ¬[F]-x1...xn)›
4137      using "RN[prem]"[where Γ="{«∀x1...∀xn [F]x1...xn»}", simplified] by blast
4138    AOT_hence ‹Impossible([F]-)›
4139      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(2)]
4140      by blast
4141  }
4142  moreover {
4143    AOT_assume ‹Impossible([F])›
4144    AOT_hence ‹□(∀x1...∀xn ¬[F]x1...xn)›
4145      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)]
4146      by blast
4147    moreover AOT_modally_strict {
4148      AOT_assume ‹∀x1...∀xn ¬[F]x1...xn›
4149      AOT_hence ‹¬[F]x1...xn› for x1xn using "∀E" by blast
4150      AOT_hence ‹[F]-x1...xn› for x1xn
4151        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(1))
4152      AOT_hence ‹∀x1...∀xn [F]-x1...xn› using "∀I" by fast
4153    }
4154    ultimately AOT_have ‹□(∀x1...∀xn [F]-x1...xn)›
4155      using "RN[prem]"[where Γ="{«∀x1...∀xn ¬[F]x1...xn»}"] by blast
4156    AOT_hence ‹Necessary([F]-)›
4157      using "≡dfI"[OF "contingent-properties:1"] by blast
4158  }
4159  ultimately AOT_have ‹Necessary([F]-) ∨ Impossible([F]-)›
4160    using "∨E"(1) "∨I" "→I" by metis
4161  AOT_thus ‹NonContingent([F]-)›
4162    using "≡dfI"[OF "contingent-properties:3"] by blast
4163next
4164  AOT_assume ‹NonContingent([F]-)›
4165  AOT_hence ‹Necessary([F]-) ∨ Impossible([F]-)›
4166    using "≡dfE"[OF "contingent-properties:3"] by blast
4167  moreover {
4168    AOT_assume ‹Necessary([F]-)›
4169    AOT_hence ‹□(∀x1...∀xn [F]-x1...xn)›
4170      using "≡dfE"[OF "contingent-properties:1"] by blast
4171    moreover AOT_modally_strict {
4172      AOT_assume ‹∀x1...∀xn [F]-x1...xn›
4173      AOT_hence ‹[F]-x1...xn› for x1xn using "∀E" by blast
4174      AOT_hence ‹¬[F]x1...xn› for x1xn
4175        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(2))
4176      AOT_hence ‹∀x1...∀xn ¬[F]x1...xn› using "∀I" by fast
4177    }
4178    ultimately AOT_have ‹□∀x1...∀xn ¬[F]x1...xn›
4179      using "RN[prem]"[where Γ="{«∀x1...∀xn [F]-x1...xn»}"] by blast
4180    AOT_hence ‹Impossible([F])›
4181      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(2)]
4182      by blast
4183  }
4184  moreover {
4185    AOT_assume ‹Impossible([F]-)›
4186    AOT_hence ‹□(∀x1...∀xn ¬[F]-x1...xn)›
4187      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(1)]
4188      by blast
4189    moreover AOT_modally_strict {
4190      AOT_assume ‹∀x1...∀xn ¬[F]-x1...xn›
4191      AOT_hence ‹¬[F]-x1...xn› for x1xn using "∀E" by blast
4192      AOT_hence ‹[F]x1...xn› for x1xn 
4193        using "thm-relation-negation:1"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4194              "useful-tautologies:1"[THEN "→E"] by blast
4195      AOT_hence ‹∀x1...∀xn [F]x1...xn› using "∀I" by fast
4196    }
4197    ultimately AOT_have ‹□(∀x1...∀xn [F]x1...xn)›
4198      using "RN[prem]"[where Γ="{«∀x1...∀xn ¬[F]-x1...xn»}"] by blast
4199    AOT_hence ‹Necessary([F])›
4200      using "≡dfI"[OF "contingent-properties:1"] by blast
4201  }
4202  ultimately AOT_have ‹Necessary([F]) ∨ Impossible([F])›
4203    using "∨E"(1) "∨I" "→I" by metis
4204  AOT_thus ‹NonContingent([F])›
4205    using "≡dfI"[OF "contingent-properties:3"] by blast
4206qed
4207
4208AOT_theorem "thm-cont-prop:2": ‹Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4209proof -
4210  AOT_have ‹Contingent([F]) ≡ ¬(Necessary([F]) ∨ Impossible([F]))›
4211    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]]
4212    by blast
4213  also AOT_have ‹... ≡ ¬Necessary([F]) & ¬Impossible([F])›
4214    using "oth-class-taut:5:d" by fastforce
4215  also AOT_have ‹... ≡ ¬Impossible([F]) & ¬Necessary([F])›
4216    by (simp add: "Commutativity of &")
4217  also AOT_have ‹... ≡ ◇∃x [F]x & ¬Necessary([F])›
4218  proof (rule "oth-class-taut:4:e"[THEN "→E"])
4219    AOT_have ‹¬Impossible([F]) ≡ ¬□¬ ∃x [F]x›
4220      apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4221      apply (AOT_subst ‹∃x [F]x› ‹¬ ∀x ¬[F]x›)
4222       apply (simp add: "conventions:4" "≡Df")
4223      apply (AOT_subst (reverse) ‹¬¬∀x ¬[F]x› ‹∀x ¬[F]x›)
4224       apply (simp add: "oth-class-taut:3:b")
4225      using "contingent-properties:2"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]] by blast
4226    also AOT_have ‹... ≡ ◇∃x [F]x›
4227      using "conventions:5"[THEN "≡Df", symmetric] by blast
4228    finally AOT_show ‹¬Impossible([F]) ≡ ◇∃x [F]x› .
4229  qed
4230  also AOT_have ‹... ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4231  proof (rule "oth-class-taut:4:f"[THEN "→E"])
4232    AOT_have ‹¬Necessary([F]) ≡ ¬□¬∃x ¬[F]x›
4233      apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4234      apply (AOT_subst ‹∃x ¬[F]x› ‹¬ ∀x ¬¬[F]x›)
4235       apply (simp add: "conventions:4" "≡Df")
4236      apply (AOT_subst (reverse) ‹¬¬[F]x› ‹[F]x› for: x)
4237       apply (simp add: "oth-class-taut:3:b")
4238      apply (AOT_subst (reverse) ‹¬¬∀x [F]x› ‹∀x [F]x›)
4239      by (auto simp: "oth-class-taut:3:b" "contingent-properties:1" "≡Df")
4240    also AOT_have ‹... ≡ ◇∃x ¬[F]x›
4241      using "conventions:5"[THEN "≡Df", symmetric] by blast
4242    finally AOT_show ‹¬Necessary([F]) ≡ ◇∃x ¬[F]x›.
4243  qed
4244  finally show ?thesis.
4245qed
4246
4247AOT_theorem "thm-cont-prop:3": ‹Contingent([F]) ≡ Contingent([F]-)› for F::‹<κ> AOT_var›
4248proof -
4249  {
4250    fix Π :: ‹<κ>›
4251    AOT_assume ‹Π↓›
4252    moreover AOT_have ‹∀F (Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x)›
4253      using "thm-cont-prop:2" GEN by fast
4254    ultimately  AOT_have ‹Contingent([Π]) ≡ ◇∃x [Π]x & ◇∃x ¬[Π]x›
4255      using "thm-cont-prop:2" "∀E" by fast
4256  } note 1 = this
4257  AOT_have ‹Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4258    using "thm-cont-prop:2" by blast
4259  also AOT_have ‹... ≡ ◇∃x ¬[F]x & ◇∃x [F]x›
4260    by (simp add: "Commutativity of &")
4261  also AOT_have ‹... ≡ ◇∃x [F]-x & ◇∃x [F]x›
4262    by (AOT_subst ‹[F]-x› ‹¬[F]x› for: x)
4263       (auto simp: "thm-relation-negation:1" "oth-class-taut:3:a")
4264  also AOT_have ‹... ≡ ◇∃x [F]-x & ◇∃x ¬[F]-x›
4265    by (AOT_subst (reverse) ‹[F]x› ‹¬[F]-x› for: x)
4266       (auto simp: "thm-relation-negation:2" "oth-class-taut:3:a")
4267  also AOT_have ‹... ≡ Contingent([F]-)›
4268    using 1[OF "rel-neg-T:3", symmetric] by blast
4269  finally show ?thesis.
4270qed
4271
4272AOT_define concrete_if_concrete :: ‹Π› ("L")  L_def: ‹L =df [λx E!x → E!x]›
4273
4274AOT_theorem "thm-noncont-e-e:1": ‹Necessary(L)›
4275proof -
4276  AOT_modally_strict {
4277    fix x
4278    AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
4279    moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
4280    moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
4281    ultimately AOT_have ‹[λx E!x → E!x]x›
4282      using "β←C" by blast
4283  }
4284  AOT_hence 0: ‹□∀x [λx E!x → E!x]x›
4285    using RN GEN by blast
4286  show ?thesis
4287    apply (rule "=dfI"(2)[OF L_def])
4288     apply "cqt:2[lambda]"
4289    by (rule "contingent-properties:1"[THEN "≡dfI", OF 0])
4290qed
4291
4292AOT_theorem "thm-noncont-e-e:2": ‹Impossible([L]-)›
4293proof -
4294  AOT_modally_strict {
4295    fix x
4296
4297    AOT_have 0: ‹∀F (¬[F]-x ≡ [F]x)›
4298      using "thm-relation-negation:2" GEN by fast
4299    AOT_have ‹¬[λx E!x → E!x]-x ≡ [λx E!x → E!x]x›
4300      by (rule 0[THEN "∀E"(1)]) "cqt:2[lambda]"
4301    moreover {
4302      AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
4303      moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
4304      moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
4305      ultimately AOT_have ‹[λx E!x → E!x]x›
4306        using "β←C" by blast
4307    }
4308    ultimately AOT_have ‹¬[λx E!x → E!x]-x›
4309      using "≡E" by blast
4310  }
4311  AOT_hence 0: ‹□∀x ¬[λx E!x → E!x]-x›
4312    using RN GEN by fast
4313  show ?thesis
4314    apply (rule "=dfI"(2)[OF L_def])
4315     apply "cqt:2[lambda]"
4316    apply (rule "contingent-properties:2"[THEN "≡dfI"]; rule "&I")
4317     using "rel-neg-T:3"
4318     apply blast
4319    using 0
4320    by blast
4321qed
4322
4323AOT_theorem "thm-noncont-e-e:3": ‹NonContingent(L)›
4324  using "thm-noncont-e-e:1"
4325  by (rule "contingent-properties:3"[THEN "≡dfI", OF "∨I"(1)])
4326
4327AOT_theorem "thm-noncont-e-e:4": ‹NonContingent([L]-)›
4328proof -
4329  AOT_have 0: ‹∀F (NonContingent([F]) ≡ NonContingent([F]-))›
4330    using "thm-cont-prop:1" "∀I" by fast
4331  moreover AOT_have 1: ‹L↓›
4332    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4333  AOT_show ‹NonContingent([L]-)›
4334    using "∀E"(1)[OF 0, OF 1, THEN "≡E"(1), OF "thm-noncont-e-e:3"] by blast
4335qed
4336
4337AOT_theorem "thm-noncont-e-e:5": ‹∃F ∃G (F ≠ «G::<κ>» & NonContingent([F]) & NonContingent([G]))›
4338proof (rule "∃I")+
4339  {
4340    AOT_have ‹∀F [F] ≠ [F]-› using "thm-relation-negation:5" GEN by fast
4341    moreover AOT_have ‹L↓›
4342      by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4343    ultimately AOT_have ‹L ≠ [L]-› using "∀E" by blast
4344  }
4345  AOT_thus ‹L ≠ [L]- & NonContingent(L) & NonContingent([L]-)›
4346    using "thm-noncont-e-e:3" "thm-noncont-e-e:4" "&I" by metis
4347next
4348  AOT_show ‹[L]-↓›
4349    using "rel-neg-T:3" by blast
4350next
4351  AOT_show ‹L↓›
4352      by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4353qed
4354
4355AOT_theorem "lem-cont-e:1": ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x)›
4356proof -
4357  AOT_have ‹◇∃x ([F]x & ◇¬[F]x) ≡ ∃x ◇([F]x & ◇¬[F]x)›
4358    using "BF◇" "CBF◇" "≡I" by blast
4359  also AOT_have ‹… ≡ ∃x (◇[F]x &  ◇¬[F]x)›
4360    by (AOT_subst ‹◇([F]x & ◇¬[F]x)› ‹◇[F]x &  ◇¬[F]x› for: x)
4361       (auto simp: "S5Basic:11" "cqt-further:7")
4362  also AOT_have ‹… ≡ ∃x (◇¬[F]x & ◇[F]x)›
4363    by (AOT_subst ‹◇¬[F]x & ◇[F]x›  ‹◇[F]x & ◇¬[F]x› for: x)
4364       (auto simp: "Commutativity of &" "cqt-further:7")
4365  also AOT_have ‹… ≡ ∃x ◇(¬[F]x & ◇[F]x)›
4366    by (AOT_subst ‹◇(¬[F]x & ◇[F]x)› ‹◇¬[F]x & ◇[F]x› for: x)
4367       (auto simp: "S5Basic:11" "oth-class-taut:3:a")
4368  also AOT_have ‹… ≡ ◇∃x (¬[F]x & ◇[F]x)›
4369    using "BF◇" "CBF◇" "≡I" by fast
4370  finally show ?thesis.
4371qed
4372
4373AOT_theorem "lem-cont-e:2": ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x ([F]-x & ◇¬[F]-x)›
4374proof -
4375  AOT_have ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x)›
4376    using "lem-cont-e:1".
4377  also AOT_have ‹… ≡ ◇∃x ([F]-x & ◇¬[F]-x)›
4378    apply (AOT_subst ‹¬[F]-x› ‹[F]x› for: x)
4379     apply (simp add: "thm-relation-negation:2")
4380    apply (AOT_subst ‹[F]-x› ‹¬[F]x› for: x)
4381     apply (simp add: "thm-relation-negation:1")
4382    by (simp add: "oth-class-taut:3:a")
4383  finally show ?thesis.
4384qed
4385
4386AOT_theorem "thm-cont-e:1": ‹◇∃x (E!x & ◇¬E!x)›
4387proof (rule "CBF◇"[THEN "→E"])
4388  AOT_have ‹∃x ◇(E!x & ¬𝒜E!x)› using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] by blast
4389  then AOT_obtain a where ‹◇(E!a & ¬𝒜E!a)› using "∃E"[rotated] by blast
4390  AOT_hence θ: ‹◇E!a & ◇¬𝒜E!a›
4391    using "KBasic2:3"[THEN "→E"] by blast
4392  AOT_have ξ: ‹◇E!a & ◇𝒜¬E!a›
4393    by (AOT_subst  ‹𝒜¬E!a› ‹¬𝒜E!a›)
4394       (auto simp: "logic-actual-nec:1"[axiom_inst] θ)
4395  AOT_have ζ: ‹◇E!a & 𝒜¬E!a›
4396    by (AOT_subst ‹𝒜¬E!a› ‹◇𝒜¬E!a›)
4397       (auto simp add: "Act-Sub:4" ξ)
4398  AOT_hence ‹◇E!a & ◇¬E!a›
4399    using "&E" "&I" "Act-Sub:3"[THEN "→E"] by blast
4400  AOT_hence ‹◇(E!a & ◇¬E!a)› using "S5Basic:11"[THEN "≡E"(2)] by simp
4401  AOT_thus ‹∃x ◇(E!x & ◇¬E!x)› using "∃I"(2) by fast
4402qed
4403
4404AOT_theorem "thm-cont-e:2": ‹◇∃x (¬E!x & ◇E!x)›
4405proof -
4406  AOT_have ‹∀F (◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x))›
4407    using "lem-cont-e:1" GEN by fast
4408  AOT_hence ‹(◇∃x (E!x & ◇¬E!x) ≡ ◇∃x (¬E!x & ◇E!x))›
4409    using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4410  thus ?thesis using "thm-cont-e:1" "≡E" by blast
4411qed
4412
4413AOT_theorem "thm-cont-e:3": ‹◇∃x E!x›
4414proof (rule "CBF◇"[THEN "→E"])
4415  AOT_obtain a where ‹◇(E!a & ◇¬E!a)›
4416    using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4417  AOT_hence ‹◇E!a›
4418    using "KBasic2:3"[THEN "→E", THEN "&E"(1)] by blast
4419  AOT_thus ‹∃x ◇E!x› using "∃I" by fast
4420qed
4421
4422AOT_theorem "thm-cont-e:4": ‹◇∃x ¬E!x›
4423proof (rule "CBF◇"[THEN "→E"])
4424  AOT_obtain a where ‹◇(E!a & ◇¬E!a)›
4425    using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4426  AOT_hence ‹◇◇¬E!a›
4427    using "KBasic2:3"[THEN "→E", THEN "&E"(2)] by blast
4428  AOT_hence ‹◇¬E!a›
4429    using "4◇"[THEN "→E"] by blast
4430  AOT_thus ‹∃x ◇¬E!x› using "∃I" by fast
4431qed
4432
4433AOT_theorem "thm-cont-e:5": ‹Contingent([E!])›
4434proof -
4435  AOT_have ‹∀F (Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x)›
4436    using "thm-cont-prop:2" GEN by fast
4437  AOT_hence ‹Contingent([E!]) ≡ ◇∃x E!x & ◇∃x ¬E!x›
4438    using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4439  thus ?thesis
4440    using "thm-cont-e:3" "thm-cont-e:4" "≡E"(2) "&I" by blast
4441qed
4442
4443AOT_theorem "thm-cont-e:6": ‹Contingent([E!]-)›
4444proof -
4445  AOT_have ‹∀F (Contingent([«F::<κ>»]) ≡ Contingent([F]-))›
4446    using "thm-cont-prop:3" GEN by fast
4447  AOT_hence ‹Contingent([E!]) ≡ Contingent([E!]-)›
4448    using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4449  thus ?thesis using "thm-cont-e:5" "≡E" by blast
4450qed
4451
4452AOT_theorem "thm-cont-e:7": ‹∃F∃G (Contingent([«F::<κ>»]) & Contingent([G]) & F ≠ G)›
4453proof (rule "∃I")+
4454  AOT_have ‹∀F [«F::<κ>»] ≠ [F]-› using "thm-relation-negation:5" GEN by fast
4455  AOT_hence ‹[E!] ≠ [E!]-›
4456    using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4457  AOT_thus ‹Contingent([E!]) & Contingent([E!]-) & [E!] ≠ [E!]-›
4458    using "thm-cont-e:5" "thm-cont-e:6" "&I" by metis
4459next
4460  AOT_show ‹E!-↓›
4461    by (fact AOT)
4462next
4463  AOT_show ‹E!↓› by (fact "cqt:2[concrete]"[axiom_inst])
4464qed
4465
4466AOT_theorem "property-facts:1": ‹NonContingent([F]) → ¬∃G (Contingent([G]) & G = F)›
4467proof (rule "→I"; rule "raa-cor:2")
4468  AOT_assume ‹NonContingent([F])›
4469  AOT_hence 1: ‹Necessary([F]) ∨ Impossible([F])›
4470    using "contingent-properties:3"[THEN "≡dfE"] by blast
4471  AOT_assume ‹∃G (Contingent([G]) & G = F)›
4472  then AOT_obtain G where ‹Contingent([G]) & G = F› using "∃E"[rotated] by blast
4473  AOT_hence ‹Contingent([F])› using "rule=E" "&E" by blast
4474  AOT_hence ‹¬(Necessary([F]) ∨ Impossible([F]))›
4475    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4476  AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) & ¬(Necessary([F]) ∨ Impossible([F]))›
4477    using 1 "&I" by blast
4478qed
4479
4480AOT_theorem "property-facts:2": ‹Contingent([F]) → ¬∃G (NonContingent([G]) & G = F)›
4481proof (rule "→I"; rule "raa-cor:2")
4482  AOT_assume ‹Contingent([F])›
4483  AOT_hence 1: ‹¬(Necessary([F]) ∨ Impossible([F]))›
4484    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4485  AOT_assume ‹∃G (NonContingent([G]) & G = F)›
4486  then AOT_obtain G where ‹NonContingent([G]) & G = F› using "∃E"[rotated] by blast
4487  AOT_hence ‹NonContingent([F])› using "rule=E" "&E" by blast
4488  AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
4489    using "contingent-properties:3"[THEN "≡dfE"] by blast
4490  AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) & ¬(Necessary([F]) ∨ Impossible([F]))›
4491    using 1 "&I" by blast
4492qed
4493
4494AOT_theorem "property-facts:3": ‹L ≠ [L]- & L ≠ E! & L ≠ E!- & [L]- ≠ [E!]- & E! ≠ [E!]-›
4495proof -
4496  AOT_have noneqI: ‹Π ≠ Π'› if ‹φ{Π}› and ‹¬φ{Π'}› for φ and Π Π' :: ‹<κ>›
4497    apply (rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
4498    using "rule=E"[where φ=φ and τ=Π and σ = Π'] that "&I" by blast
4499  AOT_have contingent_denotes: ‹Π↓› if ‹Contingent([Π])› for Π :: ‹<κ>›
4500    using that "contingent-properties:4"[THEN "≡dfE", THEN "&E"(1)] by blast
4501  AOT_have not_noncontingent_if_contingent: ‹¬NonContingent([Π])› if ‹Contingent([Π])› for Π :: ‹<κ>›
4502  proof(rule RAA(2))
4503    AOT_show ‹¬(Necessary([Π]) ∨ Impossible([Π]))›
4504      using that "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF contingent_denotes[OF that], THEN "≡E"(1)] by blast
4505  next
4506    AOT_assume ‹NonContingent([Π])›
4507    AOT_thus ‹Necessary([Π]) ∨ Impossible([Π])›
4508      using "contingent-properties:3"[THEN "≡dfE"] by blast
4509  qed
4510
4511  show ?thesis
4512  proof (safe intro!: "&I")
4513    AOT_show ‹L ≠ [L]-›
4514      apply (rule "=dfI"(2)[OF L_def])
4515       apply "cqt:2[lambda]"
4516      apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]-»"])
4517       apply (rule GEN) apply (fact AOT)
4518      by "cqt:2[lambda]"
4519  next
4520    AOT_show ‹L ≠ E!›
4521      apply (rule noneqI)
4522      using "thm-noncont-e-e:3" not_noncontingent_if_contingent[OF "thm-cont-e:5"]
4523      by auto
4524  next
4525    AOT_show ‹L ≠ E!-›
4526      apply (rule noneqI)
4527      using "thm-noncont-e-e:3" apply fast
4528      apply (rule not_noncontingent_if_contingent)
4529      apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4530      using "thm-cont-prop:3" GEN apply fast
4531      using "thm-cont-e:5" by fast+
4532  next
4533    AOT_show ‹[L]- ≠ E!-›
4534      apply (rule noneqI)
4535      using "thm-noncont-e-e:4" apply fast
4536      apply (rule not_noncontingent_if_contingent)
4537      apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4538      using "thm-cont-prop:3" GEN apply fast
4539      using "thm-cont-e:5" by fast+
4540  next
4541    AOT_show ‹E! ≠ E!-›
4542      apply (rule "=dfI"(2)[OF L_def])
4543       apply "cqt:2[lambda]"
4544      apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]-»"])
4545       apply (rule GEN) apply (fact AOT)
4546      by (fact "cqt:2[concrete]"[axiom_inst])
4547  qed
4548qed
4549
4550AOT_theorem "thm-cont-propos:1": ‹NonContingent0(p) ≡ NonContingent0(((p)-))›
4551proof(rule "≡I"; rule "→I")
4552  AOT_assume ‹NonContingent0(p)›
4553  AOT_hence ‹Necessary0(p) ∨ Impossible0(p)›
4554    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4555  moreover {
4556    AOT_assume ‹Necessary0(p)›
4557    AOT_hence 1: ‹□p› using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4558    AOT_have ‹□¬((p)-)›
4559      by (AOT_subst ‹¬((p)-)› ‹p›)
4560         (auto simp add: 1 "thm-relation-negation:4")
4561    AOT_hence ‹Impossible0(((p)-))›
4562      by (rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4563  }
4564  moreover {
4565    AOT_assume ‹Impossible0(p)›
4566    AOT_hence 1: ‹□¬p›
4567      by (rule "contingent-properties:2[zero]"[THEN "≡dfE"])
4568    AOT_have ‹□((p)-)›
4569      by (AOT_subst ‹((p)-)› ‹¬p›) 
4570         (auto simp: 1 "thm-relation-negation:3")
4571    AOT_hence ‹Necessary0(((p)-))›
4572      by (rule "contingent-properties:1[zero]"[THEN "≡dfI"])
4573  }
4574  ultimately AOT_have ‹Necessary0(((p)-)) ∨ Impossible0(((p)-))›
4575    using "∨E"(1) "∨I" "→I" by metis
4576  AOT_thus ‹NonContingent0(((p)-))›
4577    using "contingent-properties:3[zero]"[THEN "≡dfI"] by blast
4578next
4579  AOT_assume ‹NonContingent0(((p)-))›
4580  AOT_hence ‹Necessary0(((p)-)) ∨ Impossible0(((p)-))›
4581    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4582  moreover {
4583    AOT_assume ‹Impossible0(((p)-))›
4584    AOT_hence 1: ‹□¬((p)-)›
4585      by (rule "contingent-properties:2[zero]"[THEN "≡dfE"])
4586    AOT_have ‹□p›
4587      by (AOT_subst (reverse) ‹p› ‹¬((p)-)›)
4588         (auto simp: 1 "thm-relation-negation:4")
4589    AOT_hence ‹Necessary0(p)›
4590      using "contingent-properties:1[zero]"[THEN "≡dfI"] by blast
4591  }
4592  moreover {
4593    AOT_assume ‹Necessary0(((p)-))›
4594    AOT_hence 1: ‹□((p)-)›
4595      by (rule "contingent-properties:1[zero]"[THEN "≡dfE"])
4596    AOT_have ‹□¬p›
4597      by (AOT_subst (reverse) ‹¬p› ‹((p)-)›)
4598         (auto simp: 1 "thm-relation-negation:3")
4599    AOT_hence ‹Impossible0(p)›
4600      by (rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4601  }
4602  ultimately AOT_have ‹Necessary0(p) ∨ Impossible0(p)›
4603    using "∨E"(1) "∨I" "→I" by metis
4604  AOT_thus ‹NonContingent0(p)›
4605    using "contingent-properties:3[zero]"[THEN "≡dfI"] by blast
4606qed
4607
4608AOT_theorem "thm-cont-propos:2": ‹Contingent0(φ) ≡ ◇φ & ◇¬φ›
4609proof -
4610  AOT_have ‹Contingent0(φ) ≡ ¬(Necessary0(φ) ∨ Impossible0(φ))›
4611    using "contingent-properties:4[zero]"[THEN "≡Df"] by simp
4612  also AOT_have ‹… ≡ ¬Necessary0(φ) & ¬Impossible0(φ)›
4613    by (fact AOT)
4614  also AOT_have ‹… ≡ ¬Impossible0(φ) & ¬Necessary0(φ)›
4615    by (fact AOT)
4616  also AOT_have ‹… ≡ ◇φ & ◇¬φ›
4617    apply (AOT_subst ‹◇φ› ‹¬□¬φ›)
4618     apply (simp add: "conventions:5" "≡Df")
4619    apply (AOT_subst ‹Impossible0(φ)› ‹□¬φ›)
4620     apply (simp add: "contingent-properties:2[zero]" "≡Df")
4621    apply (AOT_subst (reverse) ‹◇¬φ› ‹¬□φ›)
4622     apply (simp add: "KBasic:11")
4623    apply (AOT_subst ‹Necessary0(φ)› ‹□φ›)
4624     apply (simp add: "contingent-properties:1[zero]" "≡Df")
4625    by (simp add: "oth-class-taut:3:a")
4626  finally show ?thesis.
4627qed
4628
4629AOT_theorem "thm-cont-propos:3": ‹Contingent0(p) ≡ Contingent0(((p)-))›
4630proof -
4631  AOT_have ‹Contingent0(p) ≡ ◇p & ◇¬p› using "thm-cont-propos:2".
4632  also AOT_have ‹… ≡ ◇¬p & ◇p› by (fact AOT)
4633  also AOT_have ‹… ≡ ◇((p)-) & ◇p›
4634    by (AOT_subst ‹((p)-)› ‹¬p›)
4635       (auto simp: "thm-relation-negation:3" "oth-class-taut:3:a")
4636  also AOT_have ‹… ≡ ◇((p)-) & ◇¬((p)-)›
4637    by (AOT_subst ‹¬((p)-)› ‹p›)
4638       (auto simp: "thm-relation-negation:4" "oth-class-taut:3:a")
4639  also AOT_have ‹… ≡ Contingent0(((p)-))›
4640    using "thm-cont-propos:2"[symmetric] by blast
4641  finally show ?thesis.
4642qed
4643
4644AOT_define noncontingent_prop :: ‹φ› ("p0")
4645  p0_def: "(p0) =df (∀x (E!x → E!x))"
4646
4647AOT_theorem "thm-noncont-propos:1":  ‹Necessary0((p0))›
4648proof(rule "contingent-properties:1[zero]"[THEN "≡dfI"])
4649  AOT_show ‹□(p0)›
4650    apply (rule "=dfI"(2)[OF p0_def])
4651    using "log-prop-prop:2" apply simp
4652    using "if-p-then-p" RN GEN by fast
4653qed
4654
4655AOT_theorem "thm-noncont-propos:2": ‹Impossible0(((p0)-))›
4656proof(rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4657  AOT_show ‹□¬((p0)-)›
4658    apply (AOT_subst ‹((p0)-)› ‹¬p0›)
4659    using "thm-relation-negation:3" GEN "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4660    apply (AOT_subst (reverse) ‹¬¬p0› ‹p0›)
4661     apply (simp add: "oth-class-taut:3:b")
4662    apply (rule "=dfI"(2)[OF p0_def])
4663    using "log-prop-prop:2" apply simp
4664    using "if-p-then-p" RN GEN by fast
4665qed
4666
4667AOT_theorem "thm-noncont-propos:3": ‹NonContingent0((p0))›
4668  apply(rule "contingent-properties:3[zero]"[THEN "≡dfI"])
4669  using "thm-noncont-propos:1" "∨I" by blast
4670
4671AOT_theorem "thm-noncont-propos:4": ‹NonContingent0(((p0)-))›
4672  apply(rule "contingent-properties:3[zero]"[THEN "≡dfI"])
4673  using "thm-noncont-propos:2" "∨I" by blast
4674
4675AOT_theorem "thm-noncont-propos:5": ‹∃p∃q (NonContingent0((p)) & NonContingent0((q)) & p ≠ q)›
4676proof(rule "∃I")+
4677  AOT_have 0: ‹φ ≠ (φ)-› for φ
4678    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4679  AOT_thus ‹NonContingent0((p0)) & NonContingent0(((p0)-)) & (p0) ≠ (p0)-›
4680    using "thm-noncont-propos:3" "thm-noncont-propos:4" "&I" by auto
4681qed(auto simp: "log-prop-prop:2")
4682
4683AOT_act_theorem "no-cnac": ‹¬∃x(E!x & ¬𝒜E!x)›
4684proof(rule "raa-cor:2")
4685  AOT_assume ‹∃x(E!x & ¬𝒜E!x)›
4686  then AOT_obtain a where a: ‹E!a & ¬𝒜E!a›
4687    using "∃E"[rotated] by blast
4688  AOT_hence ‹𝒜¬E!a› using "&E" "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
4689  AOT_hence ‹¬E!a› using "logic-actual"[act_axiom_inst, THEN "→E"] by blast
4690  AOT_hence ‹E!a & ¬E!a› using a "&E" "&I" by blast
4691  AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
4692qed
4693
4694AOT_theorem "pos-not-pna:1": ‹¬𝒜∃x (E!x & ¬𝒜E!x)›
4695proof(rule "raa-cor:2")
4696  AOT_assume ‹𝒜∃x (E!x & ¬𝒜E!x)›
4697  AOT_hence ‹∃x 𝒜(E!x & ¬𝒜E!x)›
4698    using "Act-Basic:10"[THEN "≡E"(1)] by blast
4699  then AOT_obtain a where ‹𝒜(E!a & ¬𝒜E!a)› using "∃E"[rotated] by blast
4700  AOT_hence 1: ‹𝒜E!a & 𝒜¬𝒜E!a› using "Act-Basic:2"[THEN "≡E"(1)] by blast
4701  AOT_hence ‹¬𝒜𝒜E!a› using "&E"(2) "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by blast
4702  AOT_hence ‹¬𝒜E!a› using "logic-actual-nec:4"[axiom_inst, THEN "≡E"(1)] RAA by blast
4703  AOT_thus ‹p & ¬p› for p using 1[THEN "&E"(1)] "&I" "raa-cor:1" by blast
4704qed
4705
4706AOT_theorem "pos-not-pna:2": ‹◇¬∃x(E!x & ¬𝒜E!x)›
4707proof (rule RAA(1))
4708  AOT_show ‹¬𝒜∃x (E!x & ¬𝒜E!x)› using "pos-not-pna:1" by blast
4709next
4710  AOT_assume ‹¬◇¬∃x (E!x & ¬𝒜E!x)›
4711  AOT_hence ‹□∃x (E!x & ¬𝒜E!x)›
4712    using "KBasic:12"[THEN "≡E"(2)] by blast
4713  AOT_thus ‹𝒜∃x (E!x & ¬𝒜E!x)›
4714    using "nec-imp-act"[THEN "→E"] by blast
4715qed
4716
4717AOT_theorem "pos-not-pna:3": ‹∃x (◇E!x & ¬𝒜E!x)›
4718proof -
4719  AOT_obtain a where ‹◇(E!a & ¬𝒜E!a)›
4720    using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] "∃E"[rotated] by blast
4721  AOT_hence θ: ‹◇E!a› and ξ: ‹◇¬𝒜E!a› using "KBasic2:3"[THEN "→E"] "&E" by blast+
4722  AOT_have ‹¬□𝒜E!a› using ξ "KBasic:11"[THEN "≡E"(2)] by blast
4723  AOT_hence ‹¬𝒜E!a› using "Act-Basic:6"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)] by blast
4724  AOT_hence ‹◇E!a & ¬𝒜E!a› using θ "&I" by blast
4725  thus ?thesis using "∃I" by fast
4726qed
4727
4728AOT_define contingent_prop :: φ ("q0")
4729  q0_def: ‹(q0) =df (∃x (E!x & ¬𝒜E!x))›
4730
4731AOT_theorem q0_prop: ‹◇q0 & ◇¬q0›
4732  apply (rule "=dfI"(2)[OF q0_def])
4733  apply (fact "log-prop-prop:2")
4734  apply (rule "&I")
4735   apply (fact "qml:4"[axiom_inst])
4736  by (fact "pos-not-pna:2")
4737
4738AOT_theorem "basic-prop:1": ‹Contingent0((q0))›
4739proof(rule "contingent-properties:4[zero]"[THEN "≡dfI"])
4740  AOT_have ‹¬Necessary0((q0)) & ¬Impossible0((q0))›
4741  proof (rule "&I"; rule "=dfI"(2)[OF q0_def]; (rule "log-prop-prop:2" | rule "raa-cor:2"))
4742    AOT_assume ‹Necessary0(∃x (E!x & ¬𝒜E!x))›
4743    AOT_hence ‹□∃x (E!x & ¬𝒜E!x)›
4744      using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4745    AOT_hence ‹𝒜∃x (E!x & ¬𝒜E!x)›
4746      using "Act-Basic:8"[THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
4747    AOT_thus ‹𝒜∃x (E!x & ¬𝒜E!x) & ¬𝒜∃x (E!x & ¬𝒜E!x)›
4748      using "pos-not-pna:1" "&I" by blast
4749  next
4750    AOT_assume ‹Impossible0(∃x (E!x & ¬𝒜E!x))›
4751    AOT_hence ‹□¬(∃x (E!x & ¬𝒜E!x))›
4752      using "contingent-properties:2[zero]"[THEN "≡dfE"] by blast
4753    AOT_hence ‹¬◇(∃x (E!x & ¬𝒜E!x))› using "KBasic2:1"[THEN "≡E"(1)] by blast
4754    AOT_thus ‹◇(∃x (E!x & ¬𝒜E!x)) & ¬◇(∃x (E!x & ¬𝒜E!x))›
4755      using "qml:4"[axiom_inst] "&I" by blast
4756  qed
4757  AOT_thus ‹¬(Necessary0((q0)) ∨ Impossible0((q0)))›
4758    using "oth-class-taut:5:d" "≡E"(2) by blast
4759qed
4760
4761AOT_theorem "basic-prop:2": ‹∃p Contingent0((p))›
4762  using "∃I"(1)[rotated, OF "log-prop-prop:2"] "basic-prop:1" by blast
4763
4764AOT_theorem "basic-prop:3": ‹Contingent0(((q0)-))›
4765  apply (AOT_subst ‹((q0)-)› ‹¬q0›)
4766   apply (insert "thm-relation-negation:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"]; fast)
4767  apply (rule "contingent-properties:4[zero]"[THEN "≡dfI"])
4768  apply (rule "oth-class-taut:5:d"[THEN "≡E"(2)])
4769  apply (rule "&I")
4770   apply (rule "contingent-properties:1[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4771   apply (rule "conventions:5"[THEN "≡dfE"])
4772   apply (rule "=dfE"(2)[OF q0_def])
4773    apply (rule "log-prop-prop:2")
4774   apply (rule q0_prop[THEN "&E"(1)])
4775  apply (rule "contingent-properties:2[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4776  apply (rule "conventions:5"[THEN "≡dfE"])
4777  by (rule q0_prop[THEN "&E"(2)])
4778
4779AOT_theorem "basic-prop:4": ‹∃p∃q (p ≠ q & Contingent0(p) & Contingent0(q))›
4780proof(rule "∃I")+
4781  AOT_have 0: ‹φ ≠ (φ)-› for φ
4782    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4783  AOT_show ‹(q0) ≠ (q0)- & Contingent0(q0) & Contingent0(((q0)-))›
4784    using "basic-prop:1" "basic-prop:3" "&I" 0 by presburger
4785qed(auto simp: "log-prop-prop:2")
4786
4787AOT_theorem "proposition-facts:1": ‹NonContingent0(p) → ¬∃q (Contingent0(q) & q = p)›
4788proof(rule "→I"; rule "raa-cor:2")
4789  AOT_assume ‹NonContingent0(p)›
4790  AOT_hence 1: ‹Necessary0(p) ∨ Impossible0(p)›
4791    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4792  AOT_assume ‹∃q (Contingent0(q) & q = p)›
4793  then AOT_obtain q where ‹Contingent0(q) & q = p› using "∃E"[rotated] by blast
4794  AOT_hence ‹Contingent0(p)› using "rule=E" "&E" by fast
4795  AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) & ¬(Necessary0(p) ∨ Impossible0(p))›
4796    using "contingent-properties:4[zero]"[THEN "≡dfE"] 1 "&I" by blast
4797qed
4798
4799AOT_theorem "proposition-facts:2": ‹Contingent0(p) → ¬∃q (NonContingent0(q) & q = p)›
4800proof(rule "→I"; rule "raa-cor:2")
4801  AOT_assume ‹Contingent0(p)›
4802  AOT_hence 1: ‹¬(Necessary0(p) ∨ Impossible0(p))›
4803    using "contingent-properties:4[zero]"[THEN "≡dfE"] by blast
4804  AOT_assume ‹∃q (NonContingent0(q) & q = p)›
4805  then AOT_obtain q where ‹NonContingent0(q) & q = p› using "∃E"[rotated] by blast
4806  AOT_hence ‹NonContingent0(p)› using "rule=E" "&E" by fast
4807  AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) & ¬(Necessary0(p) ∨ Impossible0(p))›
4808    using "contingent-properties:3[zero]"[THEN "≡dfE"] 1 "&I" by blast
4809qed
4810
4811AOT_theorem "proposition-facts:3": ‹(p0) ≠ (p0)- & (p0) ≠ (q0) & (p0) ≠ (q0)- & (p0)- ≠ (q0)- & (q0) ≠ (q0)-›
4812proof -
4813  {
4814    fix χ φ ψ
4815    AOT_assume ‹χ{φ}›
4816    moreover AOT_assume ‹¬χ{ψ}›
4817    ultimately AOT_have ‹¬(χ{φ} ≡ χ{ψ})›
4818      using RAA "≡E" by metis
4819    moreover {
4820      AOT_have ‹∀p∀q ((¬(χ{p} ≡ χ{q})) → p ≠ q)›
4821        by (rule "∀I"; rule "∀I"; rule "pos-not-equiv-ne:4[zero]")
4822      AOT_hence ‹((¬(χ{φ} ≡ χ{ψ})) → φ ≠ ψ)›
4823        using "∀E" "log-prop-prop:2" by blast
4824    }
4825    ultimately AOT_have ‹φ ≠ ψ›
4826      using "→E" by blast
4827  } note 0 = this
4828  AOT_have contingent_neg: ‹Contingent0(φ) ≡ Contingent0(((φ)-))› for φ
4829    using "thm-cont-propos:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4830  AOT_have not_noncontingent_if_contingent: ‹¬NonContingent0(φ)› if ‹Contingent0(φ)› for φ
4831    apply (rule "contingent-properties:3[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4832    using that "contingent-properties:4[zero]"[THEN "≡dfE"] by blast
4833  show ?thesis
4834    apply (rule "&I")+
4835    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4836       apply (rule 0)
4837    using "thm-noncont-propos:3" apply fast
4838       apply (rule not_noncontingent_if_contingent)
4839       apply (fact AOT)
4840      apply (rule 0)
4841    apply (rule "thm-noncont-propos:3")
4842      apply (rule not_noncontingent_if_contingent)
4843      apply (rule contingent_neg[THEN "≡E"(1)])
4844      apply (fact AOT)
4845     apply (rule 0)
4846    apply (rule "thm-noncont-propos:4")
4847      apply (rule not_noncontingent_if_contingent)
4848      apply (rule contingent_neg[THEN "≡E"(1)])
4849     apply (fact AOT)
4850    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4851qed
4852
4853AOT_define ContingentlyTrue :: ‹φ ⇒ φ› ("ContingentlyTrue'(_')")
4854  "cont-tf:1": ‹ContingentlyTrue(p) ≡df p & ◇¬p›
4855
4856AOT_define ContingentlyFalse :: ‹φ ⇒ φ› ("ContingentlyFalse'(_')")
4857  "cont-tf:2": ‹ContingentlyFalse(p) ≡df ¬p & ◇p›
4858
4859AOT_theorem "cont-true-cont:1": ‹ContingentlyTrue((p)) → Contingent0((p))›
4860proof(rule "→I")
4861  AOT_assume ‹ContingentlyTrue((p))›
4862  AOT_hence 1: ‹p› and 2: ‹◇¬p› using "cont-tf:1"[THEN "≡dfE"] "&E" by blast+
4863  AOT_have ‹¬Necessary0((p))›
4864    apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4865    using 2 "KBasic:11"[THEN "≡E"(2)] by blast
4866  moreover AOT_have ‹¬Impossible0((p))›
4867    apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4868    apply (rule "conventions:5"[THEN "≡dfE"])
4869    using "T◇"[THEN "→E", OF 1].
4870  ultimately AOT_have ‹¬(Necessary0((p)) ∨ Impossible0((p)))›
4871    using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4872  AOT_thus ‹Contingent0((p))›
4873    using "contingent-properties:4[zero]"[THEN "≡dfI"] by blast
4874qed
4875
4876AOT_theorem "cont-true-cont:2": ‹ContingentlyFalse((p)) → Contingent0((p))›
4877proof(rule "→I")
4878  AOT_assume ‹ContingentlyFalse((p))›
4879  AOT_hence 1: ‹¬p› and 2: ‹◇p› using "cont-tf:2"[THEN "≡dfE"] "&E" by blast+
4880  AOT_have ‹¬Necessary0((p))›
4881    apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4882    using "KBasic:11"[THEN "≡E"(2)] "T◇"[THEN "→E", OF 1] by blast
4883  moreover AOT_have ‹¬Impossible0((p))›
4884    apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4885    apply (rule "conventions:5"[THEN "≡dfE"])
4886    using 2.
4887  ultimately AOT_have ‹¬(Necessary0((p)) ∨ Impossible0((p)))›
4888    using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4889  AOT_thus ‹Contingent0((p))›
4890    using "contingent-properties:4[zero]"[THEN "≡dfI"] by blast
4891qed
4892
4893AOT_theorem "cont-true-cont:3": ‹ContingentlyTrue((p)) ≡ ContingentlyFalse(((p)-))›
4894proof(rule "≡I"; rule "→I")
4895  AOT_assume ‹ContingentlyTrue((p))›
4896  AOT_hence 0: ‹p & ◇¬p› using "cont-tf:1"[THEN "≡dfE"] by blast
4897  AOT_have 1: ‹ContingentlyFalse(¬p)›
4898    apply (rule "cont-tf:2"[THEN "≡dfI"])
4899    apply (AOT_subst (reverse) ‹¬¬p› p)
4900    by (auto simp: "oth-class-taut:3:b" 0)
4901  AOT_show ‹ContingentlyFalse(((p)-))›
4902    apply (AOT_subst ‹((p)-)› ‹¬p›)
4903    by (auto simp: "thm-relation-negation:3" 1)
4904next
4905  AOT_assume 1: ‹ContingentlyFalse(((p)-))›
4906  AOT_have ‹ContingentlyFalse(¬p)›
4907    by (AOT_subst (reverse) ‹¬p› ‹((p)-)›)
4908       (auto simp: "thm-relation-negation:3" 1)
4909  AOT_hence ‹¬¬p & ◇¬p› using "cont-tf:2"[THEN "≡dfE"] by blast
4910  AOT_hence ‹p & ◇¬p›
4911    using "&I" "&E" "useful-tautologies:1"[THEN "→E"] by metis
4912  AOT_thus ‹ContingentlyTrue((p))›
4913    using "cont-tf:1"[THEN "≡dfI"] by blast
4914qed
4915
4916AOT_theorem "cont-true-cont:4": ‹ContingentlyFalse((p)) ≡ ContingentlyTrue(((p)-))›
4917proof(rule "≡I"; rule "→I")
4918  AOT_assume ‹ContingentlyFalse(p)›
4919  AOT_hence 0: ‹¬p & ◇p›
4920    using "cont-tf:2"[THEN "≡dfE"] by blast
4921  AOT_have ‹¬p & ◇¬¬p›
4922    by (AOT_subst (reverse) ‹¬¬p› p)
4923       (auto simp: "oth-class-taut:3:b" 0)
4924  AOT_hence 1: ‹ContingentlyTrue(¬p)›
4925    by (rule "cont-tf:1"[THEN "≡dfI"])
4926  AOT_show ‹ContingentlyTrue(((p)-))›
4927    by (AOT_subst ‹((p)-)› ‹¬p›)
4928       (auto simp: "thm-relation-negation:3" 1)
4929next
4930  AOT_assume 1: ‹ContingentlyTrue(((p)-))›
4931  AOT_have ‹ContingentlyTrue(¬p)›
4932    by (AOT_subst (reverse) ‹¬p› ‹((p)-)›)
4933       (auto simp add: "thm-relation-negation:3" 1)
4934  AOT_hence 2: ‹¬p & ◇¬¬p› using "cont-tf:1"[THEN "≡dfE"] by blast
4935  AOT_have ‹◇p›
4936    by (AOT_subst p ‹¬¬p›)
4937       (auto simp add: "oth-class-taut:3:b" 2[THEN "&E"(2)])
4938  AOT_hence ‹¬p & ◇p› using 2[THEN "&E"(1)] "&I" by blast
4939  AOT_thus ‹ContingentlyFalse(p)›
4940    by (rule "cont-tf:2"[THEN "≡dfI"])
4941qed
4942
4943AOT_theorem "cont-true-cont:5": ‹(ContingentlyTrue((p)) & Necessary0((q))) → p ≠ q›
4944proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4945  AOT_assume ‹ContingentlyTrue((p))›
4946  AOT_hence ‹◇¬p›
4947    using "cont-tf:1"[THEN "≡dfE"] "&E" by blast
4948  AOT_hence 0: ‹¬□p› using "KBasic:11"[THEN "≡E"(2)] by blast
4949  AOT_assume ‹Necessary0((q))›
4950  moreover AOT_assume ‹¬(p ≠ q)›
4951  AOT_hence ‹p = q›
4952    using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4953          "useful-tautologies:1"[THEN "→E"] by blast
4954  ultimately AOT_have ‹Necessary0((p))› using "rule=E" id_sym by blast
4955  AOT_hence ‹□p›
4956    using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4957  AOT_thus ‹□p & ¬□p› using 0 "&I" by blast
4958qed
4959
4960AOT_theorem "cont-true-cont:6": ‹(ContingentlyFalse((p)) & Impossible0((q))) → p ≠ q›
4961proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4962  AOT_assume ‹ContingentlyFalse((p))›
4963  AOT_hence ‹◇p›
4964    using "cont-tf:2"[THEN "≡dfE"] "&E" by blast
4965  AOT_hence 1: ‹¬□¬p›
4966    using "conventions:5"[THEN "≡dfE"] by blast
4967  AOT_assume ‹Impossible0((q))›
4968  moreover AOT_assume ‹¬(p ≠ q)›
4969  AOT_hence ‹p = q›
4970    using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4971          "useful-tautologies:1"[THEN "→E"] by blast
4972  ultimately AOT_have ‹Impossible0((p))› using "rule=E" id_sym by blast
4973  AOT_hence ‹□¬p›
4974    using "contingent-properties:2[zero]"[THEN "≡dfE"] by blast
4975  AOT_thus ‹□¬p & ¬□¬p› using 1 "&I" by blast
4976qed
4977
4978AOT_act_theorem "q0cf:1": ‹ContingentlyFalse(q0)›
4979  apply (rule "cont-tf:2"[THEN "≡dfI"])
4980  apply (rule "=dfI"(2)[OF q0_def])
4981   apply (fact "log-prop-prop:2")
4982  apply (rule "&I")
4983   apply (fact "no-cnac")
4984  by (fact "qml:4"[axiom_inst])
4985
4986AOT_act_theorem "q0cf:2": ‹ContingentlyTrue(((q0)-))›
4987  apply (rule "cont-tf:1"[THEN "≡dfI"])
4988  apply (rule "=dfI"(2)[OF q0_def])
4989   apply (fact "log-prop-prop:2")
4990  apply (rule "&I")
4991     apply (rule "thm-relation-negation:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(2)])
4992     apply (fact "no-cnac")
4993    apply (rule "rule=E"[rotated, OF "thm-relation-negation:7"[unvarify p, OF "log-prop-prop:2", THEN id_sym]])
4994  apply (AOT_subst (reverse) ‹¬¬(∃x  (E!x & ¬𝒜E!x))› ‹∃x (E!x & ¬𝒜E!x)›)
4995  by (auto simp: "oth-class-taut:3:b" "qml:4"[axiom_inst])
4996
4997(* TODO: q0cf-rem skipped for now *)
4998
4999AOT_theorem "cont-tf-thm:1": ‹∃p ContingentlyTrue((p))›
5000proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
5001  AOT_assume ‹q0›
5002  AOT_hence ‹q0 & ◇¬q0› using q0_prop[THEN "&E"(2)] "&I" by blast
5003  AOT_thus ‹ContingentlyTrue(q0)›
5004    by (rule "cont-tf:1"[THEN "≡dfI"])
5005next
5006  AOT_assume ‹¬q0›
5007  AOT_hence ‹¬q0 & ◇q0› using q0_prop[THEN "&E"(1)] "&I" by blast
5008  AOT_hence ‹ContingentlyFalse(q0)›
5009    by (rule "cont-tf:2"[THEN "≡dfI"])
5010  AOT_thus ‹ContingentlyTrue(((q0)-))›
5011    by (rule "cont-true-cont:4"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
5012qed(auto simp: "log-prop-prop:2")
5013
5014
5015AOT_theorem "cont-tf-thm:2": ‹∃p ContingentlyFalse((p))›
5016proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
5017  AOT_assume ‹q0›
5018  AOT_hence ‹q0 & ◇¬q0› using q0_prop[THEN "&E"(2)] "&I" by blast
5019  AOT_hence ‹ContingentlyTrue(q0)›
5020    by (rule "cont-tf:1"[THEN "≡dfI"])
5021  AOT_thus ‹ContingentlyFalse(((q0)-))›
5022    by (rule "cont-true-cont:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
5023next
5024  AOT_assume ‹¬q0›
5025  AOT_hence ‹¬q0 & ◇q0› using q0_prop[THEN "&E"(1)] "&I" by blast
5026  AOT_thus ‹ContingentlyFalse(q0)›
5027    by (rule "cont-tf:2"[THEN "≡dfI"])
5028qed(auto simp: "log-prop-prop:2")
5029
5030(* TODO: inspect modally strict subproof involving obtained variable *)
5031AOT_theorem "property-facts1:1": ‹∃F∃x ([F]x & ◇¬[F]x)›
5032proof -
5033  fix x
5034  AOT_obtain p1 where ‹ContingentlyTrue((p1))›
5035    using "cont-tf-thm:1" "∃E"[rotated] by blast
5036  AOT_hence 1: ‹p1 & ◇¬p1› using "cont-tf:1"[THEN "≡dfE"] by blast
5037  AOT_modally_strict {
5038    AOT_have ‹for arbitrary p: ⊢□ ([λz p]x ≡ p)›
5039      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
5040    AOT_hence ‹for arbitrary p: ⊢□ □ ([λz p]x ≡ p)›
5041      by (rule RN)
5042    AOT_hence ‹∀p □([λz p]x ≡ p)› using GEN by fast
5043    AOT_hence ‹□([λz p1]x ≡ p1)› using "∀E" by fast
5044  } note 2 = this
5045  AOT_hence ‹□([λz p1]x ≡ p1)› using "∀E" by blast
5046  AOT_hence ‹[λz p1]x› using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
5047  moreover AOT_have ‹◇¬[λz p1]x›
5048    using 2[THEN "qml:2"[axiom_inst, THEN "→E"]]
5049    apply (AOT_subst ‹[λz p1]x› ‹p1›)
5050    using 1[THEN "&E"(2)] by blast
5051  ultimately AOT_have ‹[λz p1]x & ◇¬[λz p1]x› using "&I" by blast
5052  AOT_hence ‹∃x ([λz p1]x & ◇¬[λz p1]x)› using "∃I"(2) by fast
5053  moreover AOT_have ‹[λz p1]↓› by "cqt:2[lambda]"
5054  ultimately AOT_show ‹∃F∃x ([F]x & ◇¬[F]x)› by (rule "∃I"(1))
5055qed
5056
5057(* TODO: inspect modally strict subproof involving obtained variable *)
5058AOT_theorem "property-facts1:2": ‹∃F∃x (¬[F]x & ◇[F]x)›
5059proof -
5060  fix x
5061  AOT_obtain p1 where ‹ContingentlyFalse((p1))›
5062    using "cont-tf-thm:2" "∃E"[rotated] by blast
5063  AOT_hence 1: ‹¬p1 & ◇p1› using "cont-tf:2"[THEN "≡dfE"] by blast
5064  AOT_modally_strict {
5065    AOT_have ‹for arbitrary p: ⊢□ ([λz p]x ≡ p)›
5066      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
5067    AOT_hence ‹for arbitrary p: ⊢□ (¬[λz p]x ≡ ¬p)›
5068      using "oth-class-taut:4:b" "≡E" by blast
5069    AOT_hence ‹for arbitrary p: ⊢□ □(¬[λz p]x ≡ ¬p)›
5070      by (rule RN)
5071    AOT_hence ‹∀p □(¬[λz p]x ≡ ¬p)› using GEN by fast
5072    AOT_hence ‹□(¬[λz p1]x ≡ ¬p1)› using "∀E" by fast
5073  } note 2 = this
5074  AOT_hence ‹□(¬[λz p1]x ≡ ¬p1)› using "∀E" by blast
5075  AOT_hence 3: ‹¬[λz p1]x› using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
5076  AOT_modally_strict {
5077    AOT_have ‹for arbitrary p: ⊢□ ([λz p]x ≡ p)›
5078      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
5079    AOT_hence ‹for arbitrary p: ⊢□ □([λz p]x ≡ p)›
5080      by (rule RN)
5081    AOT_hence ‹∀p □([λz p]x ≡ p)› using GEN by fast
5082    AOT_hence ‹□([λz p1]x ≡ p1)› using "∀E" by fast
5083  } note 4 = this
5084  AOT_have ‹◇[λz p1]x›
5085    using 4[THEN "qml:2"[axiom_inst, THEN "→E"]]
5086    apply (AOT_subst ‹[λz p1]x› ‹p1›)
5087    using 1[THEN "&E"(2)] by blast
5088  AOT_hence ‹¬[λz p1]x & ◇[λz p1]x› using 3 "&I" by blast
5089  AOT_hence ‹∃x (¬[λz p1]x & ◇[λz p1]x)› using "∃I"(2) by fast
5090  moreover AOT_have ‹[λz p1]↓› by "cqt:2[lambda]"
5091  ultimately AOT_show ‹∃F∃x (¬[F]x & ◇[F]x)› by (rule "∃I"(1))
5092qed
5093
5094context
5095begin
5096
5097private AOT_lemma eqnotnec_123_Aux_ζ: ‹[L]x ≡ (E!x → E!x)›
5098    apply (rule "=dfI"(2)[OF L_def])
5099     apply "cqt:2[lambda]"
5100    apply (rule "beta-C-meta"[THEN "→E"])
5101  by "cqt:2[lambda]"
5102
5103private AOT_lemma eqnotnec_123_Aux_ω: ‹[λz φ]x ≡ φ›
5104    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5105
5106private AOT_lemma eqnotnec_123_Aux_θ: ‹φ ≡ ∀x([L]x ≡ [λz φ]x)›
5107proof(rule "≡I"; rule "→I"; (rule "∀I")?)
5108  fix x
5109  AOT_assume 1: ‹φ›
5110  AOT_have ‹[L]x ≡ (E!x → E!x)› using eqnotnec_123_Aux_ζ.
5111  also AOT_have ‹… ≡ φ›
5112    using "if-p-then-p" 1 "≡I" "→I" by simp
5113  also AOT_have ‹… ≡ [λz φ]x›
5114    using "Commutativity of ≡"[THEN "≡E"(1)] eqnotnec_123_Aux_ω by blast
5115  finally AOT_show ‹[L]x ≡ [λz φ]x›.
5116next
5117  fix x
5118  AOT_assume ‹∀x([L]x ≡ [λz φ]x)›
5119  AOT_hence ‹[L]x ≡ [λz φ]x› using "∀E" by blast
5120  also AOT_have ‹… ≡ φ› using eqnotnec_123_Aux_ω.
5121  finally AOT_have ‹φ ≡ [L]x› using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5122  also AOT_have ‹… ≡ E!x → E!x› using eqnotnec_123_Aux_ζ.
5123  finally AOT_show ‹φ› using "≡E" "if-p-then-p" by fast
5124qed
5125private lemmas eqnotnec_123_Aux_ξ =  eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5126                      THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5127                      THEN "RM◇"]
5128private lemmas eqnotnec_123_Aux_ξ' = eqnotnec_123_Aux_θ[THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)], THEN "RM◇"]
5129
5130AOT_theorem "eqnotnec:1": ‹∃F∃G(∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5131proof-
5132  AOT_obtain p1 where ‹ContingentlyTrue(p1)› using "cont-tf-thm:1" "∃E"[rotated] by blast
5133  AOT_hence ‹p1 & ◇¬p1› using "cont-tf:1"[THEN "≡dfE"] by blast
5134  AOT_hence ‹∀x ([L]x ≡ [λz p1]x) & ◇¬∀x([L]x ≡ [λz p1]x)›
5135    apply - apply (rule "&I")
5136    using "&E" eqnotnec_123_Aux_θ[THEN "≡E"(1)] eqnotnec_123_Aux_ξ "→E" by fast+
5137  AOT_hence ‹∃G (∀x([L]x ≡ [G]x) & ◇¬∀x([L]x ≡ [G]x))›
5138    by (rule "∃I") "cqt:2[lambda]"
5139  AOT_thus ‹∃F∃G (∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5140    apply (rule "∃I")
5141    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5142qed
5143
5144AOT_theorem "eqnotnec:2": ‹∃F∃G(¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5145proof-
5146  AOT_obtain p1 where ‹ContingentlyFalse(p1)› using "cont-tf-thm:2" "∃E"[rotated] by blast
5147  AOT_hence ‹¬p1 & ◇p1› using "cont-tf:2"[THEN "≡dfE"] by blast
5148  AOT_hence ‹¬∀x ([L]x ≡ [λz p1]x) & ◇∀x([L]x ≡ [λz p1]x)›
5149    apply - apply (rule "&I")
5150    using "&E" eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)] eqnotnec_123_Aux_ξ' "→E" by fast+
5151  AOT_hence ‹∃G (¬∀x([L]x ≡ [G]x) & ◇∀x([L]x ≡ [G]x))›
5152    by (rule "∃I") "cqt:2[lambda]"
5153  AOT_thus ‹∃F∃G (¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5154    apply (rule "∃I")
5155    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5156qed
5157
5158AOT_theorem "eqnotnec:3": ‹∃F∃G(𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5159proof-
5160  AOT_have ‹¬𝒜q0›
5161    apply (rule "=dfI"(2)[OF q0_def])
5162     apply (fact "log-prop-prop:2")
5163    by (fact AOT)
5164  AOT_hence ‹𝒜¬q0›
5165    using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5166  AOT_hence ‹𝒜¬∀x ([L]x ≡ [λz q0]x)›
5167    using eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5168            THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5169            THEN "RA[2]", THEN "act-cond"[THEN "→E"], THEN "→E"] by blast
5170  moreover AOT_have ‹◇∀x ([L]x ≡ [λz q0]x)› using eqnotnec_123_Aux_ξ'[THEN "→E"] q0_prop[THEN "&E"(1)] by blast
5171  ultimately AOT_have ‹𝒜¬∀x ([L]x ≡ [λz q0]x) & ◇∀x ([L]x ≡ [λz q0]x)› using "&I" by blast
5172  AOT_hence ‹∃G (𝒜¬∀x([L]x ≡ [G]x) & ◇∀x([L]x ≡ [G]x))›
5173    by (rule "∃I") "cqt:2[lambda]"
5174  AOT_thus ‹∃F∃G (𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5175    apply (rule "∃I")
5176    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5177qed
5178
5179end
5180
5181(* TODO[IMPORTANT]: proof of 219.4 ζ: appeal to (159.2) requires a theorem, but the result has local
5182   assumptions! *)
5183AOT_theorem "eqnotnec:4": ‹∀F∃G(∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5184proof(rule GEN)
5185  fix F
5186
5187  AOT_have Aux_A: ‹⊢□ ψ → ∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5188  proof(rule "→I"; rule GEN)
5189    AOT_modally_strict {
5190    fix x
5191    AOT_assume 0: ‹ψ›
5192    AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5193      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5194    also AOT_have ‹... ≡ [F]x›
5195      apply (rule "≡I"; rule "→I")
5196      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5197      using 0 "&I" by blast
5198    finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5199      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5200    }
5201  qed
5202
5203  AOT_have Aux_B: ‹⊢□ ψ → ∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5204  proof (rule "→I"; rule GEN)
5205    AOT_modally_strict {
5206      fix x
5207      AOT_assume 0: ‹ψ›
5208      AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5209        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5210      also AOT_have ‹... ≡ [F]x›
5211        apply (rule "≡I"; rule "→I")
5212        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5213        apply (rule "∨I"(1)) using 0 "&I" by blast
5214      finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5215        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5216    }
5217  qed
5218
5219  AOT_have Aux_C: ‹⊢□ ◇¬ψ → ◇¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5220  proof(rule "RM◇"; rule "→I"; rule "raa-cor:2")
5221  AOT_modally_strict {
5222      AOT_assume 0: ‹¬ψ›
5223      AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5224      AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5225      moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5226          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5227      moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5228        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5229      ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5230        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5231      moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5232      ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5233      AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5234    }
5235  qed
5236
5237  AOT_have Aux_D: ‹□∀z ([F]z ≡ [λz [F]z & ψ]z) → (◇¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ◇¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5238  proof (rule "→I")
5239    AOT_assume A: ‹□∀z([F]z ≡ [λz [F]z & ψ]z)›
5240    AOT_show ‹◇¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ◇¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5241    proof(rule "≡I"; rule "KBasic:13"[THEN "→E"];
5242          rule "RN[prem]"[where Γ="{«∀z([F]z ≡ [λz [F]z & ψ]z)»}", simplified];
5243          (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5244      AOT_modally_strict {
5245        AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5246        AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5247        AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5248        AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5249        AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5250        AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5251      }
5252    next
5253      AOT_modally_strict {
5254        AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5255        AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5256        AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5257        AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5258        AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5259        AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5260      }
5261    qed(auto simp: A)
5262  qed
5263
5264  AOT_obtain p1 where p1_prop: ‹p1 & ◇¬p1› using "cont-tf-thm:1" "∃E"[rotated] "cont-tf:1"[THEN "≡dfE"] by blast
5265  {
5266    AOT_assume 1: ‹□∀x([F]x ≡ [λz [F]z & p1]x)›
5267    AOT_have 2: ‹∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5268      using Aux_B[THEN "→E", OF p1_prop[THEN "&E"(1)]].
5269    AOT_have ‹◇¬∀x([λz [F]z & p1]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5270      using Aux_C[THEN "→E", OF p1_prop[THEN "&E"(2)]].
5271    AOT_hence 3: ‹◇¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5272      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5273    AOT_hence ‹∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x) & ◇¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)› using 2 "&I" by blast
5274    AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5275      by (rule "∃I"(1)) "cqt:2[lambda]"
5276  }
5277  moreover {
5278    AOT_assume 2: ‹¬□∀x([F]x ≡ [λz [F]z & p1]x)›
5279    AOT_hence ‹◇¬∀x([F]x ≡ [λz [F]z & p1]x)›
5280      using "KBasic:11"[THEN "≡E"(1)] by blast
5281    AOT_hence ‹∀x ([F]x ≡ [λz [F]z & p1]x) & ◇¬∀x([F]x ≡ [λz [F]z & p1]x)›
5282      using Aux_A[THEN "→E", OF p1_prop[THEN "&E"(1)]] "&I" by blast
5283    AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5284      by (rule "∃I"(1)) "cqt:2[lambda]"
5285  }
5286  ultimately AOT_show ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5287    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5288qed
5289
5290AOT_theorem "eqnotnec:5": ‹∀F∃G(¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5291proof(rule GEN)
5292  fix F
5293
5294  AOT_have Aux_A: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5295  proof(rule "RM◇"; rule "→I"; rule GEN)
5296    AOT_modally_strict {
5297    fix x
5298    AOT_assume 0: ‹ψ›
5299    AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5300      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5301    also AOT_have ‹... ≡ [F]x›
5302      apply (rule "≡I"; rule "→I")
5303      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5304      using 0 "&I" by blast
5305    finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5306      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5307    }
5308  qed
5309
5310  AOT_have Aux_B: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5311  proof (rule "RM◇"; rule "→I"; rule GEN)
5312    AOT_modally_strict {
5313      fix x
5314      AOT_assume 0: ‹ψ›
5315      AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5316        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5317      also AOT_have ‹... ≡ [F]x›
5318        apply (rule "≡I"; rule "→I")
5319        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5320        apply (rule "∨I"(1)) using 0 "&I" by blast
5321      finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5322        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5323    }
5324  qed
5325
5326  AOT_have Aux_C: ‹⊢□ ¬ψ → ¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5327  proof(rule "→I"; rule "raa-cor:2")
5328  AOT_modally_strict {
5329      AOT_assume 0: ‹¬ψ›
5330      AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5331      AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5332      moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5333          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5334      moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5335        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5336      ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5337        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5338      moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5339      ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5340      AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5341    }
5342  qed
5343
5344  AOT_have Aux_D: ‹∀z ([F]z ≡ [λz [F]z & ψ]z) → (¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5345  proof (rule "→I"; rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5346    AOT_modally_strict {
5347      AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5348      AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5349      AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5350      AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5351      AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5352      AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5353    }
5354  next
5355    AOT_modally_strict {
5356      AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5357      AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5358      AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5359      AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5360      AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5361      AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5362    }
5363  qed
5364
5365  AOT_obtain p1 where p1_prop: ‹¬p1 & ◇p1› using "cont-tf-thm:2" "∃E"[rotated] "cont-tf:2"[THEN "≡dfE"] by blast
5366  {
5367    AOT_assume 1: ‹∀x([F]x ≡ [λz [F]z & p1]x)›
5368    AOT_have 2: ‹◇∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5369      using Aux_B[THEN "→E", OF p1_prop[THEN "&E"(2)]].
5370    AOT_have ‹¬∀x([λz [F]z & p1]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5371      using Aux_C[THEN "→E", OF p1_prop[THEN "&E"(1)]].
5372    AOT_hence 3: ‹¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5373      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5374    AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x) & ◇∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)› using 2 "&I" by blast
5375    AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5376      by (rule "∃I"(1)) "cqt:2[lambda]"
5377  }
5378  moreover {
5379    AOT_assume 2: ‹¬∀x([F]x ≡ [λz [F]z & p1]x)›
5380    AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p1]x)›
5381      using "KBasic:11"[THEN "≡E"(1)] by blast
5382    AOT_hence ‹¬∀x ([F]x ≡ [λz [F]z & p1]x) & ◇∀x([F]x ≡ [λz [F]z & p1]x)›
5383      using Aux_A[THEN "→E", OF p1_prop[THEN "&E"(2)]] "&I" by blast
5384    AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5385      by (rule "∃I"(1)) "cqt:2[lambda]"
5386  }
5387  ultimately AOT_show ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5388    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5389qed
5390
5391AOT_theorem "eqnotnec:6": ‹∀F∃G(𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5392proof(rule GEN)
5393  fix F
5394
5395  AOT_have Aux_A: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5396  proof(rule "RM◇"; rule "→I"; rule GEN)
5397    AOT_modally_strict {
5398    fix x
5399    AOT_assume 0: ‹ψ›
5400    AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5401      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5402    also AOT_have ‹... ≡ [F]x›
5403      apply (rule "≡I"; rule "→I")
5404      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5405      using 0 "&I" by blast
5406    finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5407      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5408    }
5409  qed
5410
5411  AOT_have Aux_B: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5412  proof (rule "RM◇"; rule "→I"; rule GEN)
5413    AOT_modally_strict {
5414      fix x
5415      AOT_assume 0: ‹ψ›
5416      AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5417        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5418      also AOT_have ‹... ≡ [F]x›
5419        apply (rule "≡I"; rule "→I")
5420        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5421        apply (rule "∨I"(1)) using 0 "&I" by blast
5422      finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5423        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5424    }
5425  qed
5426
5427  AOT_have Aux_C: ‹⊢□ 𝒜¬ψ → 𝒜¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5428  proof(rule "act-cond"[THEN "→E"]; rule "RA[2]"; rule "→I"; rule "raa-cor:2")
5429  AOT_modally_strict {
5430      AOT_assume 0: ‹¬ψ›
5431      AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5432      AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5433      moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5434          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5435      moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5436        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5437      ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5438        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5439      moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5440      ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5441      AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5442    }
5443  qed
5444
5445  AOT_have Aux_D: ‹𝒜∀z ([F]z ≡ [λz [F]z & ψ]z) → (𝒜¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ 𝒜¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5446  proof (rule "→I"; rule "Act-Basic:5"[THEN "≡E"(1)])
5447    AOT_assume ‹𝒜∀z ([F]z ≡ [λz [F]z & ψ]z)›
5448    AOT_thus ‹𝒜(¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))›
5449    proof (rule "RA[3]"[where Γ="{«∀z ([F]z ≡ [λz [F]z & ψ]z)»}", simplified, rotated])
5450      AOT_modally_strict {
5451        AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5452        AOT_thus ‹¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5453          apply -
5454        proof(rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5455        AOT_modally_strict {
5456          AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5457          AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5458          AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5459          AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5460          AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5461          AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5462        }
5463      next
5464        AOT_modally_strict {
5465          AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5466          AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5467          AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5468          AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5469          AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5470          AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5471        }
5472      qed
5473      }
5474    qed
5475  qed
5476
5477  AOT_have ‹¬𝒜q0›
5478    apply (rule "=dfI"(2)[OF q0_def])
5479     apply (fact "log-prop-prop:2")
5480    by (fact AOT)
5481  AOT_hence q0_prop_1: ‹𝒜¬q0›
5482    using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5483  {
5484    AOT_assume 1: ‹𝒜∀x([F]x ≡ [λz [F]z & q0]x)›
5485    AOT_have 2: ‹◇∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x)›
5486      using Aux_B[THEN "→E", OF q0_prop[THEN "&E"(1)]].
5487    AOT_have ‹𝒜¬∀x([λz [F]z & q0]x ≡ [λz [F]z & q0 ∨ ¬q0]x)›
5488      using Aux_C[THEN "→E", OF q0_prop_1].
5489    AOT_hence 3: ‹𝒜¬∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x)›
5490      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5491    AOT_hence ‹𝒜¬∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x) & ◇∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x)› using 2 "&I" by blast
5492    AOT_hence ‹∃G (𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5493      by (rule "∃I"(1)) "cqt:2[lambda]"
5494  }
5495  moreover {
5496    AOT_assume 2: ‹¬𝒜∀x([F]x ≡ [λz [F]z & q0]x)›
5497    AOT_hence ‹𝒜¬∀x([F]x ≡ [λz [F]z & q0]x)›
5498      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5499    AOT_hence ‹𝒜¬∀x ([F]x ≡ [λz [F]z & q0]x) & ◇∀x([F]x ≡ [λz [F]z & q0]x)›
5500      using Aux_A[THEN "→E", OF q0_prop[THEN "&E"(1)]] "&I" by blast
5501    AOT_hence ‹∃G (𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5502      by (rule "∃I"(1)) "cqt:2[lambda]"
5503  }
5504  ultimately AOT_show ‹∃G (𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5505    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5506qed
5507
5508AOT_theorem "oa-contingent:1": ‹O! ≠ A!›
5509proof(rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
5510  fix x
5511  AOT_assume 1: ‹O! = A!›
5512  AOT_hence ‹[λx ◇E!x] = A!›
5513    by (rule "=dfE"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5514  AOT_hence ‹[λx ◇E!x] = [λx ¬◇E!x]›
5515    by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5516  moreover AOT_have ‹[λx ◇E!x]x ≡ ◇E!x›
5517    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5518  ultimately AOT_have ‹[λx ¬◇E!x]x ≡ ◇E!x›
5519    using "rule=E" by fast
5520  moreover AOT_have ‹[λx ¬◇E!x]x ≡ ¬◇E!x›
5521    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5522  ultimately AOT_have ‹◇E!x ≡ ¬◇E!x› using "≡E"(6) "Commutativity of ≡"[THEN "≡E"(1)] by blast
5523  AOT_thus "(◇E!x ≡ ¬◇E!x) & ¬(◇E!x ≡ ¬◇E!x)" using "oth-class-taut:3:c" "&I" by blast
5524qed
5525
5526AOT_theorem "oa-contingent:2": ‹O!x ≡ ¬A!x›
5527proof -
5528  AOT_have ‹O!x ≡ [λx ◇E!x]x›
5529    apply (rule "≡I"; rule "→I")
5530     apply (rule "=dfE"(2)[OF AOT_ordinary])
5531      apply "cqt:2[lambda]"
5532     apply argo
5533    apply (rule  "=dfI"(2)[OF AOT_ordinary])
5534     apply "cqt:2[lambda]"
5535    by argo
5536  also AOT_have ‹… ≡ ◇E!x›
5537    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5538  also AOT_have ‹… ≡ ¬¬◇E!x›
5539    using "oth-class-taut:3:b".
5540  also AOT_have ‹… ≡ ¬[λx ¬◇E!x]x›
5541    by (rule "beta-C-meta"[THEN "→E", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric]) "cqt:2[lambda]"
5542  also AOT_have ‹… ≡ ¬A!x›
5543    apply (rule "≡I"; rule "→I")
5544     apply (rule "=dfI"(2)[OF AOT_abstract])
5545      apply "cqt:2[lambda]"
5546     apply argo
5547    apply (rule "=dfE"(2)[OF AOT_abstract])
5548     apply "cqt:2[lambda]"
5549    by argo
5550  finally show ?thesis.
5551qed
5552
5553AOT_theorem "oa-contingent:3": ‹A!x ≡ ¬O!x›
5554  by (AOT_subst ‹A!x› ‹¬¬A!x›)
5555     (auto simp add: "oth-class-taut:3:b" "oa-contingent:2"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric])
5556
5557AOT_theorem "oa-contingent:4": ‹Contingent(O!)›
5558proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:1", THEN "≡E"(2)]; rule "&I")
5559  AOT_have ‹◇∃x E!x› using "thm-cont-e:3" .
5560  AOT_hence ‹∃x ◇E!x› using "BF◇"[THEN "→E"] by blast
5561  then AOT_obtain a where ‹◇E!a› using "∃E"[rotated] by blast
5562  AOT_hence ‹[λx ◇E!x]a›
5563    by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5564  AOT_hence ‹O!a›
5565    by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5566  AOT_hence ‹∃x O!x› using "∃I" by blast
5567  AOT_thus ‹◇∃x O!x› using "T◇"[THEN "→E"] by blast
5568next
5569  AOT_obtain a where ‹A!a›
5570    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5571  AOT_hence ‹¬O!a› using "oa-contingent:3"[THEN "≡E"(1)] by blast
5572  AOT_hence ‹∃x ¬O!x› using "∃I" by fast
5573  AOT_thus ‹◇∃x ¬O!x› using "T◇"[THEN "→E"] by blast
5574qed
5575
5576AOT_theorem "oa-contingent:5": ‹Contingent(A!)›
5577proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:2", THEN "≡E"(2)]; rule "&I")
5578  AOT_obtain a where ‹A!a›
5579    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5580  AOT_hence ‹∃x A!x› using "∃I" by fast
5581  AOT_thus ‹◇∃x A!x› using "T◇"[THEN "→E"] by blast
5582next
5583  AOT_have ‹◇∃x E!x› using "thm-cont-e:3" .
5584  AOT_hence ‹∃x ◇E!x› using "BF◇"[THEN "→E"] by blast
5585  then AOT_obtain a where ‹◇E!a› using "∃E"[rotated] by blast
5586  AOT_hence ‹[λx ◇E!x]a›
5587    by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5588  AOT_hence ‹O!a›
5589    by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5590  AOT_hence ‹¬A!a› using "oa-contingent:2"[THEN "≡E"(1)] by blast
5591  AOT_hence ‹∃x ¬A!x› using "∃I" by fast
5592  AOT_thus ‹◇∃x ¬A!x› using "T◇"[THEN "→E"] by blast
5593qed
5594
5595AOT_theorem "oa-contingent:7": ‹O!-x ≡ ¬A!-x›
5596proof -
5597  AOT_have ‹O!x ≡ ¬A!x›
5598    using "oa-contingent:2" by blast
5599  also AOT_have ‹… ≡ A!-x›
5600    using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:2"].
5601  finally AOT_have 1: ‹O!x ≡ A!-x›.
5602
5603  AOT_have ‹A!x ≡ ¬O!x›
5604    using "oa-contingent:3" by blast
5605  also AOT_have ‹… ≡ O!-x›
5606    using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:1"].
5607  finally AOT_have 2: ‹A!x ≡ O!-x›.
5608
5609  AOT_show ‹O!-x ≡ ¬A!-x›
5610    using 1[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "oa-contingent:3"[of _ x] 2[symmetric]
5611          "≡E"(5) by blast
5612qed
5613
5614AOT_theorem "oa-contingent:6": ‹O!- ≠ A!-›
5615proof (rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
5616  AOT_assume 1: ‹O!- = A!-›
5617  fix x
5618  AOT_have ‹A!-x ≡ O!-x›
5619    apply (rule "rule=E"[rotated, OF 1]) by (fact "oth-class-taut:3:a")
5620  AOT_hence ‹A!-x ≡ ¬A!-x›
5621    using "oa-contingent:7" "≡E" by fast
5622  AOT_thus ‹(A!-x ≡ ¬A!-x) & ¬(A!-x ≡ ¬A!-x)› using "oth-class-taut:3:c" "&I" by blast
5623qed
5624
5625AOT_theorem "oa-contingent:8": ‹Contingent(O!-)›
5626  using "thm-cont-prop:3"[unvarify F, OF "oa-exist:1", THEN "≡E"(1), OF "oa-contingent:4"].
5627
5628AOT_theorem "oa-contingent:9": ‹Contingent(A!-)›
5629  using "thm-cont-prop:3"[unvarify F, OF "oa-exist:2", THEN "≡E"(1), OF "oa-contingent:5"].
5630
5631AOT_define WeaklyContingent :: ‹Π ⇒ φ› ("WeaklyContingent'(_')")
5632  "df-cont-nec": "WeaklyContingent([F]) ≡df Contingent([F]) & ∀x (◇[F]x → □[F]x)"
5633
5634AOT_theorem "cont-nec-fact1:1": ‹WeaklyContingent([F]) ≡ WeaklyContingent([F]-)›
5635proof -
5636  AOT_have ‹WeaklyContingent([F]) ≡ Contingent([F]) & ∀x (◇[F]x → □[F]x)›
5637    using "df-cont-nec"[THEN "≡Df"] by blast
5638  also AOT_have ‹... ≡ Contingent([F]-) & ∀x (◇[F]x → □[F]x)›
5639    apply (rule "oth-class-taut:8:f"[THEN "≡E"(2)]; rule "→I")
5640    using "thm-cont-prop:3".
5641  also AOT_have ‹… ≡ Contingent([F]-) & ∀x (◇[F]-x → □[F]-x)›
5642  proof (rule "oth-class-taut:8:e"[THEN "≡E"(2)]; rule "→I"; rule "≡I"; rule "→I"; rule GEN; rule "→I")
5643    fix x
5644    AOT_assume 0: ‹∀x (◇[F]x → □[F]x)›
5645    AOT_assume 1: ‹◇[F]-x›
5646    AOT_have ‹◇¬[F]x›
5647      by (AOT_subst (reverse) ‹¬[F]x› ‹[F]-x›)
5648         (auto simp add: "thm-relation-negation:1" 1)
5649    AOT_hence 2: ‹¬□[F]x›
5650      using "KBasic:11"[THEN "≡E"(2)] by blast
5651    AOT_show ‹□[F]-x›
5652    proof (rule "raa-cor:1")
5653      AOT_assume 3: ‹¬□[F]-x›
5654      AOT_have ‹¬□¬[F]x›
5655        by (AOT_subst (reverse) ‹¬[F]x› ‹[F]-x›)
5656           (auto simp add: "thm-relation-negation:1" 3)
5657      AOT_hence ‹◇[F]x›
5658        using "conventions:5"[THEN "≡dfI"] by simp
5659      AOT_hence ‹□[F]x› using 0 "∀E" "→E" by fast
5660      AOT_thus ‹□[F]x & ¬□[F]x› using "&I" 2 by blast
5661    qed
5662  next
5663    fix x
5664    AOT_assume 0: ‹∀x (◇[F]-x → □[F]-x)›
5665    AOT_assume 1: ‹◇[F]x›
5666    AOT_have ‹◇¬[F]-x›
5667      by (AOT_subst ‹¬[F]-x› ‹[F]x›)
5668         (auto simp: "thm-relation-negation:2" 1)
5669    AOT_hence 2: ‹¬□[F]-x›
5670      using "KBasic:11"[THEN "≡E"(2)] by blast
5671    AOT_show ‹□[F]x›
5672    proof (rule "raa-cor:1")
5673      AOT_assume 3: ‹¬□[F]x›
5674      AOT_have ‹¬□¬[F]-x›
5675        by (AOT_subst ‹¬[F]-x› ‹[F]x›)
5676           (auto simp add: "thm-relation-negation:2" 3)
5677      AOT_hence ‹◇[F]-x›
5678        using "conventions:5"[THEN "≡dfI"] by simp
5679      AOT_hence ‹□[F]-x› using 0 "∀E" "→E" by fast
5680      AOT_thus ‹□[F]-x & ¬□[F]-x› using "&I" 2 by blast
5681    qed
5682  qed
5683  also AOT_have ‹… ≡ WeaklyContingent([F]-)›
5684    using "df-cont-nec"[THEN "≡Df", symmetric] by blast
5685  finally show ?thesis.
5686qed
5687
5688AOT_theorem "cont-nec-fact1:2": ‹(WeaklyContingent([F]) & ¬WeaklyContingent([G])) → F ≠ G›
5689proof (rule "→I"; rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
5690  AOT_assume 1: ‹WeaklyContingent([F]) & ¬WeaklyContingent([G])›
5691  AOT_hence ‹WeaklyContingent([F])› using "&E" by blast
5692  moreover AOT_assume ‹F = G›
5693  ultimately AOT_have ‹WeaklyContingent([G])›
5694    using "rule=E" by blast
5695  AOT_thus ‹WeaklyContingent([G]) & ¬WeaklyContingent([G])›
5696    using 1 "&I" "&E" by blast
5697qed
5698
5699AOT_theorem "cont-nec-fact2:1": ‹WeaklyContingent(O!)›
5700proof (rule "df-cont-nec"[THEN "≡dfI"]; rule "&I")
5701  AOT_show ‹Contingent(O!)›
5702    using "oa-contingent:4".
5703next
5704  AOT_show ‹∀x (◇[O!]x → □[O!]x)›
5705    apply (rule GEN; rule "→I")
5706    using "oa-facts:5"[THEN "≡E"(1)] by blast
5707qed
5708
5709
5710AOT_theorem "cont-nec-fact2:2": ‹WeaklyContingent(A!)›
5711proof (rule "df-cont-nec"[THEN "≡dfI"]; rule "&I")
5712  AOT_show ‹Contingent(A!)›
5713    using "oa-contingent:5".
5714next
5715  AOT_show ‹∀x (◇[A!]x → □[A!]x)›
5716    apply (rule GEN; rule "→I")
5717    using "oa-facts:6"[THEN "≡E"(1)] by blast
5718qed
5719
5720AOT_theorem "cont-nec-fact2:3": ‹¬WeaklyContingent(E!)›
5721proof (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5722       rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "raa-cor:2")
5723  AOT_have ‹◇∃x (E!x & ¬𝒜E!x)› using "qml:4"[axiom_inst].
5724  AOT_hence ‹∃x ◇(E!x & ¬𝒜E!x)› using "BF◇"[THEN "→E"] by blast
5725  then AOT_obtain a where ‹◇(E!a & ¬𝒜E!a)› using "∃E"[rotated] by blast
5726  AOT_hence 1: ‹◇E!a & ◇¬𝒜E!a› using "KBasic2:3"[THEN "→E"] by simp
5727  moreover AOT_assume ‹∀x (◇[E!]x → □[E!]x)›
5728  ultimately AOT_have ‹□E!a› using "&E" "∀E" "→E" by fast
5729  AOT_hence ‹𝒜E!a› using "nec-imp-act"[THEN "→E"] by blast
5730  AOT_hence ‹□𝒜E!a› using "qml-act:1"[axiom_inst, THEN "→E"] by blast
5731  moreover AOT_have ‹¬□𝒜E!a› using "KBasic:11"[THEN "≡E"(2)] 1[THEN "&E"(2)] by meson
5732  ultimately AOT_have ‹□𝒜E!a & ¬□𝒜E!a› using "&I" by blast
5733  AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
5734qed
5735
5736AOT_theorem "cont-nec-fact2:4": ‹¬WeaklyContingent(L)›
5737  apply (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5738       rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(1))
5739  apply (rule "contingent-properties:4"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
5740  apply (rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "useful-tautologies:2"[THEN "→E"])
5741  using "thm-noncont-e-e:3"[THEN "contingent-properties:3"[THEN "≡dfE"]].
5742
5743(* TODO: cleanup *)
5744AOT_theorem "cont-nec-fact2:5": ‹O! ≠ E! & O! ≠ E!- & O! ≠ L & O! ≠ L-›
5745proof -
5746  AOT_have 1: ‹L↓›
5747    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5748  {
5749    fix φ and Π Π' :: ‹<κ>›
5750    AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if  ‹φ{Π}› and ‹¬φ{Π'}›
5751    proof (rule "raa-cor:2")
5752      AOT_assume ‹φ{Π'} ≡ φ{Π}›
5753      AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
5754      AOT_thus ‹φ{Π'} & ¬φ{Π'}› using that(2) "&I" by blast
5755    qed
5756    AOT_have ‹Π' ≠ Π› if ‹Π↓› and ‹Π'↓› and ‹φ{Π}› and ‹¬φ{Π'}›
5757      using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5758  } note 0 = this
5759  show ?thesis
5760    apply(safe intro!: "&I"; rule 0)
5761    using "cqt:2[concrete]"[axiom_inst] apply blast
5762    using "oa-exist:1" apply blast
5763    using "cont-nec-fact2:3" apply fast
5764    apply (rule "useful-tautologies:2"[THEN "→E"])
5765    using "cont-nec-fact2:1" apply fast
5766    using "rel-neg-T:3" apply fast
5767    using "oa-exist:1" apply blast
5768    using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5769    apply (rule "useful-tautologies:2"[THEN "→E"])
5770    using "cont-nec-fact2:1" apply blast
5771    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5772    using "oa-exist:1" apply fast
5773    using "cont-nec-fact2:4" apply fast
5774    apply (rule "useful-tautologies:2"[THEN "→E"])
5775    using "cont-nec-fact2:1" apply fast
5776    using "rel-neg-T:3" apply fast
5777    using "oa-exist:1" apply fast
5778    apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5779    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5780    apply (rule "useful-tautologies:2"[THEN "→E"])
5781    using "cont-nec-fact2:1" by blast
5782qed
5783
5784(* TODO: cleanup together with above *)
5785AOT_theorem "cont-nec-fact2:6": ‹A! ≠ E! & A! ≠ E!- & A! ≠ L & A! ≠ L-›
5786proof -
5787  AOT_have 1: ‹L↓›
5788    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5789  {
5790    fix φ and Π Π' :: ‹<κ>›
5791    AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if  ‹φ{Π}› and ‹¬φ{Π'}›
5792    proof (rule "raa-cor:2")
5793      AOT_assume ‹φ{Π'} ≡ φ{Π}›
5794      AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
5795      AOT_thus ‹φ{Π'} & ¬φ{Π'}› using that(2) "&I" by blast
5796    qed
5797    AOT_have ‹Π' ≠ Π› if ‹Π↓› and ‹Π'↓› and ‹φ{Π}› and ‹¬φ{Π'}›
5798      using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5799  } note 0 = this
5800  show ?thesis
5801    apply(safe intro!: "&I"; rule 0)
5802    using "cqt:2[concrete]"[axiom_inst] apply blast
5803    using "oa-exist:2" apply blast
5804    using "cont-nec-fact2:3" apply fast
5805    apply (rule "useful-tautologies:2"[THEN "→E"])
5806    using "cont-nec-fact2:2" apply fast
5807    using "rel-neg-T:3" apply fast
5808    using "oa-exist:2" apply blast
5809    using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5810    apply (rule "useful-tautologies:2"[THEN "→E"])
5811    using "cont-nec-fact2:2" apply blast
5812    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5813    using "oa-exist:2" apply fast
5814    using "cont-nec-fact2:4" apply fast
5815    apply (rule "useful-tautologies:2"[THEN "→E"])
5816    using "cont-nec-fact2:2" apply fast
5817    using "rel-neg-T:3" apply fast
5818    using "oa-exist:2" apply fast
5819    apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5820    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5821    apply (rule "useful-tautologies:2"[THEN "→E"])
5822    using "cont-nec-fact2:2" by blast
5823qed
5824
5825AOT_define necessary_or_contingently_false :: ‹φ ⇒ φ› ("Δ_" [49] 54)
5826  ‹Δp ≡df □p ∨ (¬𝒜p & ◇p)›
5827
5828AOT_theorem sixteen:
5829 shows ‹∃F1∃F2∃F3∃F4∃F5∃F6∃F7∃F8∃F9∃F10∃F11∃F12∃F13∃F14∃F15∃F16 (
5830«F1::<κ>» ≠ F2 & F1 ≠ F3 & F1 ≠ F4 & F1 ≠ F5 & F1 ≠ F6 & F1 ≠ F7 & F1 ≠ F8 & F1 ≠ F9 & F1 ≠ F10 & F1 ≠ F11 & F1 ≠ F12 & F1 ≠ F13 & F1 ≠ F14 & F1 ≠ F15 & F1 ≠ F16 &
5831F2 ≠ F3 & F2 ≠ F4 & F2 ≠ F5 & F2 ≠ F6 & F2 ≠ F7 & F2 ≠ F8 & F2 ≠ F9 & F2 ≠ F10 & F2 ≠ F11 & F2 ≠ F12 & F2 ≠ F13 & F2 ≠ F14 & F2 ≠ F15 & F2 ≠ F16 &
5832F3 ≠ F4 & F3 ≠ F5 & F3 ≠ F6 & F3 ≠ F7 & F3 ≠ F8 & F3 ≠ F9 & F3 ≠ F10 & F3 ≠ F11 & F3 ≠ F12 & F3 ≠ F13 & F3 ≠ F14 & F3 ≠ F15 & F3 ≠ F16 &
5833F4 ≠ F5 & F4 ≠ F6 & F4 ≠ F7 & F4 ≠ F8 & F4 ≠ F9 & F4 ≠ F10 & F4 ≠ F11 & F4 ≠ F12 & F4 ≠ F13 & F4 ≠ F14 & F4 ≠ F15 & F4 ≠ F16 &
5834F5 ≠ F6 & F5 ≠ F7 & F5 ≠ F8 & F5 ≠ F9 & F5 ≠ F10 & F5 ≠ F11 & F5 ≠ F12 & F5 ≠ F13 & F5 ≠ F14 & F5 ≠ F15 & F5 ≠ F16 &
5835F6 ≠ F7 & F6 ≠ F8 & F6 ≠ F9 & F6 ≠ F10 & F6 ≠ F11 & F6 ≠ F12 & F6 ≠ F13 & F6 ≠ F14 & F6 ≠ F15 & F6 ≠ F16 &
5836F7 ≠ F8 & F7 ≠ F9 & F7 ≠ F10 & F7 ≠ F11 & F7 ≠ F12 & F7 ≠ F13 & F7 ≠ F14 & F7 ≠ F15 & F7 ≠ F16 &
5837F8 ≠ F9 & F8 ≠ F10 & F8 ≠ F11 & F8 ≠ F12 & F8 ≠ F13 & F8 ≠ F14 & F8 ≠ F15 & F8 ≠ F16 &
5838F9 ≠ F10 & F9 ≠ F11 & F9 ≠ F12 & F9 ≠ F13 & F9 ≠ F14 & F9 ≠ F15 & F9 ≠ F16 &
5839F10 ≠ F11 & F10 ≠ F12 & F10 ≠ F13 & F10 ≠ F14 & F10 ≠ F15 & F10 ≠ F16 &
5840F11 ≠ F12 & F11 ≠ F13 & F11 ≠ F14 & F11 ≠ F15 & F11 ≠ F16 &
5841F12 ≠ F13 & F12 ≠ F14 & F12 ≠ F15 & F12 ≠ F16 &
5842F13 ≠ F14 & F13 ≠ F15 & F13 ≠ F16 &
5843F14 ≠ F15 & F14 ≠ F16 &
5844F15 ≠ F16)› 
5845proof -
5846
5847  AOT_have Delta_pos: ‹Δφ → ◇φ› for φ
5848  proof(rule "→I")
5849    AOT_assume ‹Δφ›
5850    AOT_hence ‹□φ ∨ (¬𝒜φ & ◇φ)›
5851      using "≡dfE"[OF necessary_or_contingently_false] by blast
5852    moreover {
5853      AOT_assume ‹□φ›
5854      AOT_hence ‹◇φ›
5855        by (metis "B◇" "T◇" "vdash-properties:10")
5856    }
5857    moreover {
5858      AOT_assume ‹¬𝒜φ & ◇φ›
5859      AOT_hence ‹◇φ›
5860        using "&E" by blast
5861    }
5862    ultimately AOT_show ‹◇φ›
5863      by (metis "∨E"(2) "raa-cor:1") 
5864  qed
5865
5866  AOT_have act_and_not_nec_not_delta: ‹¬Δφ› if ‹𝒜φ› and ‹¬□φ› for φ
5867    using "≡dfE" "&E"(1) "∨E"(2) necessary_or_contingently_false "raa-cor:3" that(1) that(2) by blast
5868  AOT_have act_and_pos_not_not_delta: ‹¬Δφ› if ‹𝒜φ› and ‹◇¬φ› for φ
5869    using "KBasic:11" act_and_not_nec_not_delta "≡E"(2) that(1) that(2) by blast
5870  AOT_have impossible_delta: ‹¬Δφ› if ‹¬◇φ› for φ
5871    using Delta_pos "modus-tollens:1" that by blast
5872  AOT_have not_act_and_pos_delta: ‹Δφ› if ‹¬𝒜φ› and ‹◇φ› for φ
5873    by (meson "≡dfI" "&I" "∨I"(2) necessary_or_contingently_false that(1) that(2))
5874  AOT_have nec_delta: ‹Δφ› if ‹□φ› for φ
5875    using "≡dfI" "∨I"(1) necessary_or_contingently_false that by blast
5876
5877  AOT_obtain a where a_prop: ‹A!a›
5878    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5879  AOT_obtain b where b_prop: ‹◇[E!]b & ¬𝒜[E!]b›
5880    using "pos-not-pna:3" using "∃E"[rotated] by blast
5881
5882  AOT_have b_ord: ‹[O!]b›
5883  proof(rule "=dfI"(2)[OF AOT_ordinary])
5884    AOT_show ‹[λx ◇[E!]x]↓› by "cqt:2[lambda]"
5885  next
5886    AOT_show ‹[λx ◇[E!]x]b›
5887    proof (rule "β←C"(1); ("cqt:2[lambda]")?)
5888      AOT_show ‹b↓› by (rule "cqt:2[const_var]"[axiom_inst])
5889      AOT_show ‹◇[E!]b› by (fact b_prop[THEN "&E"(1)])
5890    qed
5891  qed
5892
5893  AOT_have nec_not_L_neg: ‹□¬[L-]x› for x
5894    using "thm-noncont-e-e:2" "contingent-properties:2"[THEN "≡dfE"] "&E"
5895          CBF[THEN "→E"] "∀E" by blast
5896  AOT_have nec_L: ‹□[L]x› for x
5897    using "thm-noncont-e-e:1" "contingent-properties:1"[THEN "≡dfE"]
5898      CBF[THEN "→E"] "∀E" by blast
5899
5900  AOT_have act_ord_b: ‹𝒜[O!]b›
5901    using b_ord "≡E"(1) "oa-facts:7" by blast
5902  AOT_have delta_ord_b: ‹Δ[O!]b›
5903    by (meson "≡dfI" b_ord "∨I"(1) necessary_or_contingently_false "oa-facts:1" "vdash-properties:10")
5904  AOT_have not_act_ord_a: ‹¬𝒜[O!]a›
5905    by (meson a_prop "≡E"(1) "≡E"(3) "oa-contingent:3" "oa-facts:7")
5906  AOT_have not_delta_ord_a: ‹¬Δ[O!]a›
5907    by (metis Delta_pos "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "reductio-aa:1" "vdash-properties:10")
5908
5909  AOT_have not_act_abs_b: ‹¬𝒜[A!]b›
5910    by (meson b_ord "≡E"(1) "≡E"(3) "oa-contingent:2" "oa-facts:8")
5911  AOT_have not_delta_abs_b: ‹¬Δ[A!]b›
5912  proof(rule "raa-cor:2")
5913    AOT_assume ‹Δ[A!]b›
5914    AOT_hence ‹◇[A!]b›
5915      by (metis Delta_pos "vdash-properties:10")
5916    AOT_thus ‹[A!]b & ¬[A!]b›
5917      by (metis b_ord "&I" "≡E"(1) "oa-contingent:2" "oa-facts:4" "vdash-properties:10")
5918  qed
5919  AOT_have act_abs_a: ‹𝒜[A!]a›
5920    using a_prop "≡E"(1) "oa-facts:8" by blast
5921  AOT_have delta_abs_a: ‹Δ[A!]a›
5922      by (metis "≡dfI" a_prop "oa-facts:2" "vdash-properties:10" "∨I"(1) necessary_or_contingently_false)
5923
5924  AOT_have not_act_concrete_b: ‹¬𝒜[E!]b›
5925    using b_prop "&E"(2) by blast
5926  AOT_have delta_concrete_b: ‹Δ[E!]b›
5927  proof (rule "≡dfI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5928    AOT_show ‹¬𝒜[E!]b› using b_prop "&E"(2) by blast
5929  next
5930    AOT_show ‹◇[E!]b› using b_prop "&E"(1) by blast
5931  qed
5932  AOT_have not_act_concrete_a: ‹¬𝒜[E!]a›
5933  proof (rule "raa-cor:2")
5934    AOT_assume ‹𝒜[E!]a›
5935    AOT_hence 1: ‹◇[E!]a› by (metis "Act-Sub:3" "vdash-properties:10")
5936    AOT_have ‹[A!]a› by (simp add: a_prop)
5937    AOT_hence ‹[λx ¬◇[E!]x]a›
5938      by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5939    AOT_hence ‹¬◇[E!]a› using "β→C"(1) by blast
5940    AOT_thus ‹◇[E!]a & ¬◇[E!]a› using 1 "&I" by blast
5941  qed
5942  AOT_have not_delta_concrete_a: ‹¬Δ[E!]a›
5943  proof (rule "raa-cor:2")
5944    AOT_assume ‹Δ[E!]a›
5945    AOT_hence 1: ‹◇[E!]a› by (metis Delta_pos "vdash-properties:10")
5946    AOT_have ‹[A!]a› by (simp add: a_prop)
5947    AOT_hence ‹[λx ¬◇[E!]x]a›
5948      by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5949    AOT_hence ‹¬◇[E!]a› using "β→C"(1) by blast
5950    AOT_thus ‹◇[E!]a & ¬◇[E!]a› using 1 "&I" by blast
5951  qed
5952
5953  AOT_have not_act_q_zero: ‹¬𝒜q0›
5954    by (meson "log-prop-prop:2" "pos-not-pna:1" q0_def "reductio-aa:1" "rule-id-df:2:a[zero]")
5955  AOT_have delta_q_zero: ‹Δq0›
5956  proof(rule "≡dfI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5957    AOT_show ‹¬𝒜q0› using not_act_q_zero.
5958    AOT_show ‹◇q0› by (meson "&E"(1) q0_prop)
5959  qed
5960  AOT_have act_not_q_zero: ‹𝒜¬q0› using "Act-Basic:1" "∨E"(2) not_act_q_zero by blast
5961  AOT_have not_delta_not_q_zero: ‹¬Δ¬q0›
5962      using "≡dfE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta "&E"(1) "∨E"(2) not_act_q_zero q0_prop by blast
5963
5964  AOT_have ‹[L-]↓› by (simp add: "rel-neg-T:3")
5965  moreover AOT_have ‹¬𝒜[L-]b & ¬Δ[L-]b & ¬𝒜[L-]a & ¬Δ[L-]a›
5966  proof (safe intro!: "&I")
5967    AOT_show ‹¬𝒜[L-]b› by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5968    AOT_show ‹¬Δ[L-]b› by (meson Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg)
5969    AOT_show ‹¬𝒜[L-]a› by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5970    AOT_show ‹¬Δ[L-]a› using Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg by blast
5971  qed
5972  ultimately AOT_obtain F0 where ‹¬𝒜[F0]b & ¬Δ[F0]b & ¬𝒜[F0]a & ¬Δ[F0]a›
5973    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5974  AOT_hence ‹¬𝒜[F0]b› and ‹¬Δ[F0]b› and ‹¬𝒜[F0]a› and ‹¬Δ[F0]a›
5975    using "&E" by blast+
5976  note props = this
5977
5978  let ?Π = "«[λy [A!]y & q0]»"
5979  AOT_modally_strict {
5980    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
5981  } note 1 = this
5982  moreover AOT_have ‹¬𝒜[«?Π»]b & ¬Δ[«?Π»]b & ¬𝒜[«?Π»]a & Δ[«?Π»]a›
5983  proof (safe intro!: "&I"; AOT_subst ‹[λy A!y & q0]x› ‹A!x & q0› for: x)
5984    AOT_show ‹¬𝒜([A!]b & q0)›
5985      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
5986  next AOT_show ‹¬Δ([A!]b & q0)›
5987      by (metis Delta_pos "KBasic2:3" "&E"(1) "≡E"(4) not_act_abs_b "oa-facts:4" "oa-facts:8" "raa-cor:3" "vdash-properties:10")
5988  next AOT_show ‹¬𝒜([A!]a & q0)›
5989      using "Act-Basic:2" "&E"(2) "≡E"(1) not_act_q_zero "raa-cor:3" by blast
5990  next AOT_show ‹Δ([A!]a & q0)›
5991    proof (rule not_act_and_pos_delta)
5992      AOT_show ‹¬𝒜([A!]a & q0)›
5993        using "Act-Basic:2" "&E"(2) "≡E"(4) not_act_q_zero "raa-cor:3" by blast
5994    next AOT_show ‹◇([A!]a & q0)›
5995        by (metis "&I" "→E" Delta_pos "KBasic:16" "&E"(1) delta_abs_a "≡E"(1) "oa-facts:6" q0_prop)
5996    qed
5997  qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
5998  ultimately AOT_obtain F1 where ‹¬𝒜[F1]b & ¬Δ[F1]b & ¬𝒜[F1]a & Δ[F1]a›
5999    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6000  AOT_hence ‹¬𝒜[F1]b› and ‹¬Δ[F1]b› and ‹¬𝒜[F1]a› and ‹Δ[F1]a›
6001    using "&E" by blast+
6002  note props = props this
6003
6004  let ?Π = "«[λy [A!]y & ¬q0]»"
6005  AOT_modally_strict {
6006    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6007  } note 1 = this
6008  moreover AOT_have ‹¬𝒜[«?Π»]b & ¬Δ[«?Π»]b & 𝒜[«?Π»]a & ¬Δ[«?Π»]a›
6009  proof (safe intro!: "&I"; AOT_subst ‹[λy A!y & ¬q0]x› ‹A!x & ¬q0› for: x)
6010    AOT_show ‹¬𝒜([A!]b & ¬q0)›
6011      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
6012  next AOT_show ‹¬Δ([A!]b & ¬q0)›
6013      by (meson "RM◇" Delta_pos "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b "oa-facts:4" "oa-facts:8")
6014  next AOT_show ‹𝒜([A!]a & ¬q0)›
6015      by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
6016  next AOT_show ‹¬Δ([A!]a & ¬q0)›
6017    proof (rule act_and_not_nec_not_delta)
6018      AOT_show ‹𝒜([A!]a & ¬q0)›
6019        by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
6020    next
6021      AOT_show ‹¬□([A!]a & ¬q0)›
6022        by (metis "KBasic2:1" "KBasic:3" "&E"(1) "&E"(2) "≡E"(4) q0_prop "raa-cor:3")
6023    qed
6024  qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
6025  ultimately AOT_obtain F2 where ‹¬𝒜[F2]b & ¬Δ[F2]b & 𝒜[F2]a & ¬Δ[F2]a›
6026    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6027  AOT_hence ‹¬𝒜[F2]b› and ‹¬Δ[F2]b› and ‹𝒜[F2]a› and ‹¬Δ[F2]a›
6028    using "&E" by blast+
6029  note props = props this
6030
6031  AOT_have abstract_prop: ‹¬𝒜[A!]b & ¬Δ[A!]b & 𝒜[A!]a & Δ[A!]a›
6032    using act_abs_a "&I" delta_abs_a not_act_abs_b not_delta_abs_b by presburger
6033  then AOT_obtain F3 where ‹¬𝒜[F3]b & ¬Δ[F3]b & 𝒜[F3]a & Δ[F3]a›
6034    using "∃I"(1)[rotated, THEN "∃E"[rotated]] "oa-exist:2" by fastforce
6035  AOT_hence ‹¬𝒜[F3]b› and ‹¬Δ[F3]b› and ‹𝒜[F3]a› and ‹Δ[F3]a›
6036    using "&E" by blast+
6037  note props = props this
6038
6039  AOT_have ‹¬𝒜[E!]b & Δ[E!]b & ¬𝒜[E!]a & ¬Δ[E!]a›
6040    by (meson "&I" delta_concrete_b not_act_concrete_a not_act_concrete_b not_delta_concrete_a)
6041  then AOT_obtain F4 where ‹¬𝒜[F4]b & Δ[F4]b & ¬𝒜[F4]a & ¬Δ[F4]a›
6042    using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6043  AOT_hence ‹¬𝒜[F4]b› and ‹Δ[F4]b› and ‹¬𝒜[F4]a› and ‹¬Δ[F4]a›
6044    using "&E" by blast+
6045  note props = props this
6046
6047  AOT_modally_strict {
6048    AOT_have ‹[λy q0]↓› by "cqt:2[lambda]"
6049  } note 1 = this
6050  moreover AOT_have ‹¬𝒜[λy q0]b & Δ[λy q0]b & ¬𝒜[λy q0]a & Δ[λy q0]a›
6051    by (safe intro!: "&I"; AOT_subst ‹[λy q0]b› ‹q0› for: b)
6052       (auto simp: not_act_q_zero delta_q_zero  "beta-C-meta"[THEN "→E", OF 1])
6053  ultimately AOT_obtain F5 where ‹¬𝒜[F5]b & Δ[F5]b & ¬𝒜[F5]a & Δ[F5]a›
6054    using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6055  AOT_hence ‹¬𝒜[F5]b› and ‹Δ[F5]b› and ‹¬𝒜[F5]a› and ‹Δ[F5]a›
6056    using "&E" by blast+
6057  note props = props this
6058
6059  let ?Π = "«[λy [E!]y ∨ ([A!]y & ¬q0)]»"
6060  AOT_modally_strict {
6061    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6062  } note 1 = this
6063  moreover AOT_have ‹¬𝒜[«?Π»]b & Δ[«?Π»]b & 𝒜[«?Π»]a & ¬Δ[«?Π»]a›
6064  proof(safe intro!: "&I"; AOT_subst ‹[λy E!y ∨ (A!y & ¬q0)]x› ‹E!x ∨ (A!x & ¬q0)› for: x)
6065    AOT_have ‹𝒜¬([A!]b & ¬q0)›
6066      by (metis "Act-Basic:1" "Act-Basic:2" abstract_prop "&E"(1) "∨E"(2)
6067                "≡E"(1) "raa-cor:3")
6068    moreover AOT_have ‹¬𝒜[E!]b›
6069      using b_prop "&E"(2) by blast
6070    ultimately AOT_have 2: ‹𝒜(¬[E!]b & ¬([A!]b & ¬q0))›
6071      by (metis "Act-Basic:2" "Act-Sub:1" "&I" "≡E"(3) "raa-cor:1")
6072    AOT_have ‹𝒜¬([E!]b ∨ ([A!]b & ¬q0))›
6073      by (AOT_subst ‹¬([E!]b ∨ ([A!]b & ¬q0))› ‹¬[E!]b & ¬([A!]b & ¬q0)›)
6074         (auto simp: "oth-class-taut:5:d" 2)
6075    AOT_thus ‹¬𝒜([E!]b ∨ ([A!]b & ¬q0))›
6076      by (metis "¬¬I" "Act-Sub:1" "≡E"(4))
6077  next
6078    AOT_show ‹Δ([E!]b ∨ ([A!]b & ¬q0))›
6079    proof (rule not_act_and_pos_delta)
6080      AOT_show ‹¬𝒜([E!]b ∨ ([A!]b & ¬q0))›
6081        by (metis "Act-Basic:2" "Act-Basic:9" "∨E"(2) "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b not_act_concrete_b "raa-cor:3")
6082    next
6083      AOT_show ‹◇([E!]b ∨ ([A!]b & ¬q0))›
6084        using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
6085    qed
6086  next AOT_show ‹𝒜([E!]a ∨ ([A!]a & ¬q0))›
6087      by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
6088  next AOT_show ‹¬Δ([E!]a ∨ ([A!]a & ¬q0))›
6089    proof (rule act_and_not_nec_not_delta)
6090      AOT_show ‹𝒜([E!]a ∨ ([A!]a & ¬q0))›
6091        by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
6092    next
6093      AOT_have ‹□¬[E!]a›
6094        by (metis "≡dfI" "conventions:5" "&I" "∨I"(2) necessary_or_contingently_false not_act_concrete_a not_delta_concrete_a "raa-cor:3")
6095      moreover AOT_have ‹◇¬([A!]a & ¬q0)›
6096        by (metis "KBasic2:1" "KBasic:11" "KBasic:3" "&E"(1) "&E"(2) "≡E"(1) q0_prop "raa-cor:3")
6097      ultimately AOT_have ‹◇(¬[E!]a & ¬([A!]a & ¬q0))› by (metis "KBasic:16" "&I" "vdash-properties:10")
6098      AOT_hence ‹◇¬([E!]a ∨ ([A!]a & ¬q0))›
6099        by (metis "RE◇" "≡E"(2) "oth-class-taut:5:d")
6100      AOT_thus ‹¬□([E!]a ∨ ([A!]a & ¬q0))› by (metis "KBasic:12" "≡E"(1) "raa-cor:3")
6101    qed
6102  qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
6103  ultimately AOT_obtain F6 where ‹¬𝒜[F6]b & Δ[F6]b & 𝒜[F6]a & ¬Δ[F6]a›
6104    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6105  AOT_hence ‹¬𝒜[F6]b› and ‹Δ[F6]b› and ‹𝒜[F6]a› and ‹¬Δ[F6]a›
6106    using "&E" by blast+
6107  note props = props this
6108
6109  let ?Π = "«[λy [A!]y ∨ [E!]y]»"
6110  AOT_modally_strict {
6111    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6112  } note 1 = this
6113  moreover AOT_have ‹¬𝒜[«?Π»]b & Δ[«?Π»]b & 𝒜[«?Π»]a & Δ[«?Π»]a›
6114  proof(safe intro!: "&I"; AOT_subst ‹[λy A!y ∨ E!y]x› ‹A!x ∨ E!x› for: x)
6115    AOT_show ‹¬𝒜([A!]b ∨ [E!]b)›
6116      using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6117  next AOT_show ‹Δ([A!]b ∨ [E!]b)›
6118    proof (rule not_act_and_pos_delta)
6119      AOT_show ‹¬𝒜([A!]b ∨ [E!]b)›
6120        using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6121    next AOT_show ‹◇([A!]b ∨ [E!]b)›
6122        using "KBasic2:2" b_prop "&E"(1) "∨I"(2) "≡E"(2) by blast
6123    qed
6124  next AOT_show ‹𝒜([A!]a ∨ [E!]a)›
6125      by (meson "Act-Basic:9" act_abs_a "∨I"(1) "≡E"(2))
6126  next AOT_show ‹Δ([A!]a ∨ [E!]a) ›
6127    proof (rule nec_delta)
6128      AOT_show ‹□([A!]a ∨ [E!]a)›
6129        by (metis "KBasic:15" act_abs_a act_and_not_nec_not_delta "Disjunction Addition"(1) delta_abs_a "raa-cor:3" "vdash-properties:10")
6130    qed
6131  qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
6132  ultimately AOT_obtain F7 where ‹¬𝒜[F7]b & Δ[F7]b & 𝒜[F7]a & Δ[F7]a›
6133    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6134  AOT_hence ‹¬𝒜[F7]b› and ‹Δ[F7]b› and ‹𝒜[F7]a› and ‹Δ[F7]a›
6135    using "&E" by blast+
6136  note props = props this
6137
6138  let ?Π = "«[λy [O!]y & ¬[E!]y]»"
6139  AOT_modally_strict {
6140    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6141  } note 1 = this
6142  moreover AOT_have ‹𝒜[«?Π»]b & ¬Δ[«?Π»]b & ¬𝒜[«?Π»]a & ¬Δ[«?Π»]a›
6143  proof(safe intro!: "&I"; AOT_subst ‹[λy O!y & ¬E!y]x› ‹O!x & ¬E!x› for: x)
6144    AOT_show ‹𝒜([O!]b & ¬[E!]b)›
6145      by (metis "Act-Basic:1" "Act-Basic:2" act_ord_b "&I" "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:3")
6146  next AOT_show ‹¬Δ([O!]b & ¬[E!]b)›
6147      by (metis (no_types, hide_lams) "conventions:5" "Act-Sub:1" "RM:1" act_and_not_nec_not_delta "act-conj-act:3"
6148                act_ord_b b_prop "&I" "&E"(1) "Conjunction Simplification"(2) "df-rules-formulas[3]"
6149                "≡E"(3) "raa-cor:1" "→E")
6150  next AOT_show ‹¬𝒜([O!]a & ¬[E!]a)›
6151      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_ord_a "raa-cor:3" by blast
6152  next AOT_have ‹¬◇([O!]a & ¬[E!]a)›
6153      by (metis "KBasic2:3" "&E"(1) "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "raa-cor:3" "vdash-properties:10")
6154    AOT_thus ‹¬Δ([O!]a & ¬[E!]a)›
6155      by (rule impossible_delta)
6156  qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
6157  ultimately AOT_obtain F8 where ‹𝒜[F8]b & ¬Δ[F8]b & ¬𝒜[F8]a & ¬Δ[F8]a›
6158    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6159  AOT_hence ‹𝒜[F8]b› and ‹¬Δ[F8]b› and ‹¬𝒜[F8]a› and ‹¬Δ[F8]a›
6160    using "&E" by blast+
6161  note props = props this
6162
6163  (* TODO_PLM: binary property 9 wrong in PLM *)
6164  let ?Π = "«[λy ¬[E!]y & ([O!]y ∨ q0)]»"
6165  AOT_modally_strict {
6166    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6167  } note 1 = this
6168  moreover AOT_have ‹𝒜[«?Π»]b & ¬Δ[«?Π»]b & ¬𝒜[«?Π»]a & Δ[«?Π»]a›
6169  proof(safe intro!: "&I"; AOT_subst ‹[λy ¬E!y & (O!y ∨ q0)]x› ‹¬E!x & (O!x ∨ q0)› for: x)
6170    AOT_show ‹𝒜(¬[E!]b & ([O!]b ∨ q0))›
6171      by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6172                "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6173  next AOT_show ‹¬Δ(¬[E!]b & ([O!]b ∨ q0))›
6174    proof (rule act_and_pos_not_not_delta)
6175      AOT_show ‹𝒜(¬[E!]b & ([O!]b ∨ q0))›
6176        by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6177                  "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6178    next
6179      AOT_show ‹◇¬(¬[E!]b & ([O!]b ∨ q0))›
6180      proof (AOT_subst ‹¬(¬[E!]b & ([O!]b ∨ q0))› ‹[E!]b ∨ ¬([O!]b ∨ q0)›)
6181        AOT_modally_strict {
6182          AOT_show ‹¬(¬[E!]b & ([O!]b ∨ q0)) ≡ [E!]b ∨ ¬([O!]b ∨ q0)›
6183            by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2) "∨E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
6184        }
6185      next
6186        AOT_show ‹◇([E!]b ∨ ¬([O!]b ∨ q0))›
6187          using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
6188       qed
6189     qed
6190   next
6191     AOT_show ‹¬𝒜(¬[E!]a & ([O!]a ∨ q0))›
6192       using "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2" by blast
6193   next
6194     AOT_show ‹Δ(¬[E!]a & ([O!]a ∨ q0))›
6195     proof (rule not_act_and_pos_delta)
6196       AOT_show ‹¬𝒜(¬[E!]a & ([O!]a ∨ q0))›
6197         by (metis "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2")
6198     next
6199       AOT_have ‹□¬[E!]a›
6200         using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "raa-cor:5" by blast
6201       moreover AOT_have ‹◇([O!]a ∨ q0)›
6202         by (metis "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(3) q0_prop "raa-cor:3")
6203       ultimately AOT_show ‹◇(¬[E!]a & ([O!]a ∨ q0))›
6204         by (metis "KBasic:16" "&I" "vdash-properties:10")
6205     qed
6206   qed(auto simp:  "beta-C-meta"[THEN "→E", OF 1])
6207  ultimately AOT_obtain F9 where ‹𝒜[F9]b & ¬Δ[F9]b & ¬𝒜[F9]a & Δ[F9]a›
6208    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6209  AOT_hence ‹𝒜[F9]b› and ‹¬Δ[F9]b› and ‹¬𝒜[F9]a› and ‹Δ[F9]a›
6210    using "&E" by blast+
6211  note props = props this
6212
6213  AOT_modally_strict {
6214    AOT_have ‹[λy ¬q0]↓› by "cqt:2[lambda]"
6215  } note 1 = this
6216  moreover AOT_have ‹𝒜[λy ¬q0]b & ¬Δ[λy ¬q0]b & 𝒜[λy ¬q0]a & ¬Δ[λy ¬q0]a›
6217    by (safe intro!: "&I"; AOT_subst ‹[λy ¬q0]x› ‹¬q0› for: x)
6218       (auto simp: act_not_q_zero not_delta_not_q_zero "beta-C-meta"[THEN "→E", OF 1])
6219  ultimately AOT_obtain F10 where ‹𝒜[F10]b & ¬Δ[F10]b & 𝒜[F10]a & ¬Δ[F10]a›
6220    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6221  AOT_hence ‹𝒜[F10]b› and ‹¬Δ[F10]b› and ‹𝒜[F10]a› and ‹¬Δ[F10]a›
6222    using "&E" by blast+
6223  note props = props this
6224
6225  AOT_modally_strict {
6226    AOT_have ‹[λy ¬[E!]y]↓› by "cqt:2[lambda]"
6227  } note 1 = this
6228  moreover AOT_have ‹𝒜[λy ¬[E!]y]b & ¬Δ[λy ¬[E!]y]b & 𝒜[λy ¬[E!]y]a & Δ[λy ¬[E!]y]a›
6229  proof (safe intro!: "&I"; AOT_subst ‹[λy ¬[E!]y]x› ‹¬[E!]x› for: x)
6230    AOT_show ‹𝒜¬[E!]b›
6231      using "Act-Basic:1" "∨E"(2) not_act_concrete_b by blast
6232  next AOT_show ‹¬Δ¬[E!]b›
6233      using "≡dfE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta b_prop "&E"(1) "∨E"(2) not_act_concrete_b by blast
6234  next AOT_show ‹𝒜¬[E!]a›
6235      using "Act-Basic:1" "∨E"(2) not_act_concrete_a by blast
6236  next AOT_show ‹Δ¬[E!]a›
6237      using "KBasic2:1" "≡E"(2) nec_delta not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "reductio-aa:1" by blast
6238  qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
6239  ultimately AOT_obtain F11 where ‹𝒜[F11]b & ¬Δ[F11]b & 𝒜[F11]a & Δ[F11]a›
6240    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6241  AOT_hence ‹𝒜[F11]b› and ‹¬Δ[F11]b› and ‹𝒜[F11]a› and ‹Δ[F11]a›
6242    using "&E" by blast+
6243  note props = props this
6244
6245  AOT_have ‹𝒜[O!]b & Δ[O!]b & ¬𝒜[O!]a & ¬Δ[O!]a›
6246    by (simp add: act_ord_b "&I" delta_ord_b not_act_ord_a not_delta_ord_a)
6247  then AOT_obtain F12 where ‹𝒜[F12]b & Δ[F12]b & ¬𝒜[F12]a & ¬Δ[F12]a›
6248    using "oa-exist:1" "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6249  AOT_hence ‹𝒜[F12]b› and ‹Δ[F12]b› and ‹¬𝒜[F12]a› and ‹¬Δ[F12]a›
6250    using "&E" by blast+
6251  note props = props this
6252
6253  let ?Π = "«[λy [O!]y ∨ q0]»"
6254  AOT_modally_strict {
6255    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6256  } note 1 = this
6257  moreover AOT_have ‹𝒜[«?Π»]b & Δ[«?Π»]b & ¬𝒜[«?Π»]a & Δ[«?Π»]a›
6258  proof (safe intro!: "&I"; AOT_subst ‹[λy O!y ∨ q0]x› ‹O!x ∨ q0› for: x)
6259    AOT_show ‹𝒜([O!]b ∨ q0)›
6260      by (meson "Act-Basic:9" act_ord_b "∨I"(1) "≡E"(2))
6261  next AOT_show ‹Δ([O!]b ∨ q0)›
6262      by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6263  next AOT_show ‹¬𝒜([O!]a ∨ q0)›
6264      using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6265  next AOT_show ‹Δ([O!]a ∨ q0)›
6266    proof (rule not_act_and_pos_delta)
6267      AOT_show ‹¬𝒜([O!]a ∨ q0)›
6268        using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6269    next AOT_show ‹◇([O!]a ∨ q0)›
6270        using "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(2) q0_prop by blast
6271    qed
6272  qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
6273  ultimately AOT_obtain F13 where ‹𝒜[F13]b & Δ[F13]b & ¬𝒜[F13]a & Δ[F13]a›
6274    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6275  AOT_hence ‹𝒜[F13]b› and ‹Δ[F13]b› and ‹¬𝒜[F13]a› and ‹Δ[F13]a›
6276    using "&E" by blast+
6277  note props = props this
6278
6279  let ?Π = "«[λy [O!]y ∨ ¬q0]»"
6280  AOT_modally_strict {
6281     AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6282  } note 1 = this
6283  moreover AOT_have ‹𝒜[«?Π»]b & Δ[«?Π»]b & 𝒜[«?Π»]a & ¬Δ[«?Π»]a›
6284  proof (safe intro!: "&I"; AOT_subst ‹[λy O!y ∨ ¬q0]x› ‹O!x ∨ ¬q0› for: x)
6285    AOT_show ‹𝒜([O!]b ∨ ¬q0)›
6286      by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6287  next AOT_show ‹Δ([O!]b ∨ ¬q0)›
6288      by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6289  next AOT_show ‹𝒜([O!]a ∨ ¬q0)›
6290      by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6291  next AOT_show ‹¬Δ([O!]a ∨ ¬q0)›
6292    proof(rule act_and_pos_not_not_delta)
6293      AOT_show ‹𝒜([O!]a ∨ ¬q0)›
6294        by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6295    next
6296      AOT_have ‹□¬[O!]a›
6297        using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_ord_a not_delta_ord_a "raa-cor:6" by blast
6298      moreover AOT_have ‹◇q0›
6299        by (meson "&E"(1) q0_prop)
6300      ultimately AOT_have 2: ‹◇(¬[O!]a & q0)›
6301         by (metis "KBasic:16" "&I" "vdash-properties:10")
6302      AOT_show ‹◇¬([O!]a ∨ ¬q0)›
6303      proof (AOT_subst (reverse) ‹¬([O!]a ∨ ¬q0)› ‹¬[O!]a & q0›)
6304        AOT_modally_strict {
6305          AOT_show ‹¬[O!]a & q0 ≡ ¬([O!]a ∨ ¬q0)›
6306            by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2)
6307                      "∨E"(3) "deduction-theorem" "≡I" "raa-cor:3")
6308        }
6309      next
6310        AOT_show ‹◇(¬[O!]a & q0)›
6311          using "2" by blast
6312      qed
6313    qed
6314  qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
6315  ultimately AOT_obtain F14 where ‹𝒜[F14]b & Δ[F14]b & 𝒜[F14]a & ¬Δ[F14]a›
6316    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6317  AOT_hence ‹𝒜[F14]b› and ‹Δ[F14]b› and ‹𝒜[F14]a› and ‹¬Δ[F14]a›
6318    using "&E" by blast+
6319  note props = props this
6320
6321  AOT_have ‹[L]↓›
6322    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
6323  moreover AOT_have ‹𝒜[L]b & Δ[L]b & 𝒜[L]a & Δ[L]a›
6324  proof (safe intro!: "&I")
6325    AOT_show ‹𝒜[L]b›
6326      by (meson nec_L "nec-imp-act" "vdash-properties:10")
6327    next AOT_show ‹Δ[L]b› using nec_L nec_delta by blast
6328    next AOT_show ‹𝒜[L]a› by (meson nec_L "nec-imp-act" "vdash-properties:10")
6329    next AOT_show ‹Δ[L]a› using nec_L nec_delta by blast
6330  qed
6331  ultimately AOT_obtain F15 where ‹𝒜[F15]b & Δ[F15]b & 𝒜[F15]a & Δ[F15]a›
6332    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6333  AOT_hence ‹𝒜[F15]b› and ‹Δ[F15]b› and ‹𝒜[F15]a› and ‹Δ[F15]a›
6334    using "&E" by blast+
6335  note props = props this
6336
6337  show ?thesis
6338    by (rule "∃I"(2)[where β=F0]; rule "∃I"(2)[where β=F1]; rule "∃I"(2)[where β=F2];
6339           rule "∃I"(2)[where β=F3]; rule "∃I"(2)[where β=F4]; rule "∃I"(2)[where β=F5];
6340           rule "∃I"(2)[where β=F6]; rule "∃I"(2)[where β=F7]; rule "∃I"(2)[where β=F8];
6341           rule "∃I"(2)[where β=F9]; rule "∃I"(2)[where β=F10]; rule "∃I"(2)[where β=F11];
6342           rule "∃I"(2)[where β=F12]; rule "∃I"(2)[where β=F13]; rule "∃I"(2)[where β=F14];
6343           rule "∃I"(2)[where β=F15]; safe intro!: "&I")
6344       (match conclusion in "[?v ⊨ [F] ≠ [G]]" for F G ⇒ ‹
6345        match props in A: "[?v ⊨ ¬φ{F}]" for φ ⇒ ‹
6346        match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail› ¦ _ ⇒ ‹
6347        match props in B: "[?v ⊨ φ{G}]" ⇒ ‹
6348        fact "pos-not-equiv-ne:4"[where F=F and G=G and φ=φ, THEN "→E",
6349                                OF "oth-class-taut:4:h"[THEN "≡E"(2)],
6350                                OF "Disjunction Addition"(2)[THEN "→E"],
6351                                OF "&I", OF A, OF B]››››)+
6352qed
6353
6354AOT_theorem "o-objects-exist:1": ‹□∃x O!x›
6355proof(rule RN)
6356  AOT_modally_strict {
6357    AOT_obtain a where ‹◇(E!a & ¬𝒜[E!]a)›
6358      using "∃E"[rotated, OF "qml:4"[axiom_inst, THEN "BF◇"[THEN "→E"]]] by blast
6359    AOT_hence 1: ‹◇E!a› by (metis "KBasic2:3" "&E"(1) "→E")
6360    AOT_have ‹[λx ◇[E!]x]a›
6361    proof (rule "β←C"(1); "cqt:2[lambda]"?)
6362      AOT_show ‹a↓› using "cqt:2[const_var]"[axiom_inst] by blast
6363    next
6364      AOT_show ‹◇E!a› by (fact 1)
6365    qed
6366    AOT_hence ‹O!a› by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6367    AOT_thus ‹∃x [O!]x› by (rule "∃I")
6368  }
6369qed
6370
6371AOT_theorem "o-objects-exist:2": ‹□∃x A!x›
6372proof (rule RN)
6373  AOT_modally_strict {
6374    AOT_obtain a where ‹[A!]a›
6375      using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
6376    AOT_thus ‹∃x A!x› using "∃I" by blast
6377  }
6378qed
6379
6380AOT_theorem "o-objects-exist:3": ‹□¬∀x O!x›
6381  by (rule RN) (metis (no_types, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(4) "modus-tollens:1" "o-objects-exist:2" "oa-contingent:2" "qml:2"[axiom_inst] "reductio-aa:2")
6382
6383AOT_theorem "o-objects-exist:4": ‹□¬∀x A!x›
6384  by (rule RN) (metis (mono_tags, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(1) "modus-tollens:1" "o-objects-exist:1" "oa-contingent:2" "qml:2"[axiom_inst] "→E")
6385
6386AOT_theorem "o-objects-exist:5": ‹□¬∀x E!x›
6387proof (rule RN; rule "raa-cor:2")
6388  AOT_modally_strict {
6389    AOT_assume ‹∀x E!x›
6390    moreover AOT_obtain a where abs: ‹A!a›
6391      using "o-objects-exist:2"[THEN "qml:2"[axiom_inst, THEN "→E"]] "∃E"[rotated] by blast
6392    ultimately AOT_have ‹E!a› using "∀E" by blast
6393    AOT_hence 1: ‹◇E!a› by (metis "T◇" "→E")
6394    AOT_have ‹[λy ◇E!y]a›
6395    proof (rule "β←C"(1); "cqt:2[lambda]"?)
6396      AOT_show ‹a↓› using "cqt:2[const_var]"[axiom_inst].
6397    next
6398      AOT_show ‹◇E!a› by (fact 1)
6399    qed
6400    AOT_hence ‹O!a›
6401      by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6402    AOT_hence ‹¬A!a› by (metis "≡E"(1) "oa-contingent:2") 
6403    AOT_thus ‹p & ¬p› for p using abs by (metis "raa-cor:3")
6404  }
6405qed
6406
6407AOT_theorem partition: ‹¬∃x (O!x & A!x)›
6408proof(rule "raa-cor:2")
6409  AOT_assume ‹∃x (O!x & A!x)›
6410  then AOT_obtain a where ‹O!a & A!a› using "∃E"[rotated] by blast
6411  AOT_thus ‹p & ¬p› for p by (metis "&E"(1) "Conjunction Simplification"(2) "≡E"(1) "modus-tollens:1" "oa-contingent:2" "raa-cor:3")
6412qed
6413
6414AOT_define eq_E :: ‹Π› ("'(=E')") "=E": ‹(=E) =df [λxy O!x & O!y & □∀F ([F]x ≡ [F]y)]›
6415
6416syntax "_AOT_eq_E_infix" :: ‹τ ⇒ τ ⇒ φ› (infixl "=E" 50)
6417translations
6418  "_AOT_eq_E_infix κ κ'" == "CONST AOT_exe (CONST eq_E) (CONST Pair κ κ')"
6419(* TODO: try to replace by a simple translations pattern *)
6420print_translation‹
6421AOT_syntax_print_translations
6422[(const_syntax‹AOT_exe›, fn ctxt => fn [
6423  Const ("constAOT_PLM.eq_E", _),
6424  Const (const_syntax‹Pair›, _) $ lhs $ rhs
6425] => Const (syntax_const‹_AOT_eq_E_infix›, dummyT) $ lhs $ rhs)]›
6426
6427text‹Note: Not explicitly mentioned as theorem in PLM.›
6428AOT_theorem "=E[denotes]": ‹[(=E)]↓›
6429  by (rule "=dfI"(2)[OF "=E"]) "cqt:2[lambda]"+
6430
6431AOT_theorem "=E-simple:1": ‹x =E y ≡ (O!x & O!y & □∀F ([F]x ≡ [F]y))›
6432proof -
6433  (* TODO: rethink the product hacks *)
6434  AOT_have 0: ‹«(AOT_term_of_var x,AOT_term_of_var y)»↓›
6435    by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6436  AOT_have 1: ‹[λxy [O!]x & [O!]y & □∀F ([F]x ≡ [F]y)]↓› by "cqt:2[lambda]"
6437  show ?thesis apply (rule "=dfI"(2)[OF "=E"]; "cqt:2[lambda]"?)
6438    using "beta-C-meta"[THEN "→E", OF 1, unvarify ν1νn, of "(AOT_term_of_var x,AOT_term_of_var y)", OF 0]
6439    by fast
6440qed
6441
6442AOT_theorem "=E-simple:2": ‹x =E y → x = y›
6443proof (rule "→I")
6444  AOT_assume ‹x =E y›
6445  AOT_hence ‹O!x & O!y & □∀F ([F]x ≡ [F]y)› using "=E-simple:1"[THEN "≡E"(1)] by blast
6446  AOT_thus ‹x = y›
6447    using "≡dfI"[OF "identity:1"] "∨I" by blast
6448qed
6449
6450AOT_theorem "id-nec3:1": ‹x =E y ≡ □(x =E y)›
6451proof (rule "≡I"; rule "→I")
6452  AOT_assume ‹x =E y›
6453  AOT_hence ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›
6454    using "=E-simple:1" "≡E" by blast
6455  AOT_hence ‹□O!x & □O!y & □□∀F ([F]x ≡ [F]y)›
6456    by (metis "S5Basic:6" "&I" "&E"(1) "&E"(2) "≡E"(4) "oa-facts:1" "raa-cor:3" "vdash-properties:10")
6457  AOT_hence ‹□(O!x & O!y & □∀F ([F]x ≡ [F]y))›
6458    by (metis "&E"(1) "&E"(2) "≡E"(2) "KBasic:3" "&I")
6459  AOT_thus ‹□(x =E y)›
6460    using "=E-simple:1"
6461    by (AOT_subst ‹x =E y› ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›) auto
6462next
6463  AOT_assume ‹□(x =E y)›
6464  AOT_thus ‹x =E y› using "qml:2"[axiom_inst, THEN "→E"] by blast
6465qed
6466
6467AOT_theorem "id-nec3:2": ‹◇(x =E y) ≡ x =E y›
6468  by (meson "RE◇" "S5Basic:2" "id-nec3:1" "≡E"(1) "≡E"(5) "Commutativity of ≡")
6469
6470AOT_theorem "id-nec3:3": ‹◇(x =E y) ≡ □(x =E y)›
6471  by (meson "id-nec3:1" "id-nec3:2" "≡E"(5))
6472
6473syntax "_AOT_non_eq_E" :: ‹Π› ("'(≠E')")
6474translations
6475  (Π) "(≠E)" == (Π) "(=E)-"
6476syntax "_AOT_non_eq_E_infix" :: ‹τ ⇒ τ ⇒ φ› (infixl "≠E" 50)
6477translations
6478 "_AOT_non_eq_E_infix κ κ'" == "CONST AOT_exe (CONST relation_negation (CONST eq_E)) (CONST Pair κ κ')"
6479(* TODO: try replacing be a simple translations pattern *)
6480print_translation‹
6481AOT_syntax_print_translations
6482[(const_syntax‹AOT_exe›, fn ctxt => fn [
6483  Const (const_syntax‹relation_negation›, _) $ Const ("constAOT_PLM.eq_E", _),
6484  Const (const_syntax‹Pair›, _) $ lhs $ rhs
6485] => Const (syntax_const‹_AOT_non_eq_E_infix›, dummyT) $ lhs $ rhs)]›
6486AOT_theorem "thm-neg=E": ‹x ≠E y ≡ ¬(x =E y)›
6487proof -
6488  (* TODO: rethink the product hacks *)
6489  AOT_have 0: ‹«(AOT_term_of_var x,AOT_term_of_var y)»↓›
6490    by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6491  AOT_have θ: ‹[λx1...x2 ¬(=E)x1...x2]↓› by "cqt:2[lambda]" (* TODO_PLM: convoluted proof in PLM; TODO: product hack *)
6492  AOT_have ‹x ≠E y ≡ [λx1...x2 ¬(=E)x1...x2]xy›
6493    by (rule "=dfI"(1)[OF "df-relation-negation", OF θ])
6494       (meson "oth-class-taut:3:a")
6495  also AOT_have ‹… ≡ ¬(=E)xy›
6496    apply (rule "beta-C-meta"[THEN "→E", unvarify ν1νn])
6497     apply "cqt:2[lambda]"
6498    by (fact 0)
6499  finally show ?thesis.
6500qed
6501
6502AOT_theorem "id-nec4:1": ‹x ≠E y ≡ □(x ≠E y)›
6503proof -
6504  AOT_have ‹x ≠E y ≡ ¬(x =E y)› using "thm-neg=E".
6505  also AOT_have ‹… ≡ ¬◇(x =E y)›
6506    by (meson "id-nec3:2" "≡E"(1) "Commutativity of ≡" "oth-class-taut:4:b")
6507  also AOT_have ‹… ≡ □¬(x =E y)›
6508    by (meson "KBasic2:1" "≡E"(2) "Commutativity of ≡")
6509  also AOT_have ‹… ≡ □(x ≠E y)›
6510    by (AOT_subst (reverse) ‹¬(x =E y)› ‹x ≠E y›)
6511       (auto simp: "thm-neg=E" "oth-class-taut:3:a")
6512  finally show ?thesis.
6513qed
6514
6515AOT_theorem "id-nec4:2": ‹◇(x ≠E y) ≡ (x ≠E y)›
6516  by (meson "RE◇" "S5Basic:2" "id-nec4:1" "≡E"(2) "≡E"(5) "Commutativity of ≡")
6517
6518AOT_theorem "id-nec4:3": ‹◇(x ≠E y) ≡ □(x ≠E y)›
6519  by (meson "id-nec4:1" "id-nec4:2" "≡E"(5))
6520
6521AOT_theorem "id-act2:1": ‹x =E y ≡ 𝒜x =E y›
6522  by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec3:2" "≡E"(1) "≡E"(6))
6523AOT_theorem "id-act2:2": ‹x ≠E y ≡ 𝒜x ≠E y›
6524  by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec4:2" "≡E"(1) "≡E"(6))
6525
6526AOT_theorem "ord=Eequiv:1": ‹O!x → x =E x›
6527proof (rule "→I")
6528  AOT_assume 1: ‹O!x›
6529  AOT_show ‹x =E x›
6530    apply (rule "=dfI"(2)[OF "=E"]) apply "cqt:2[lambda]"
6531    apply (rule "β←C"(1))
6532      apply "cqt:2[lambda]"
6533     apply (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6534    by (simp add: "1" RN "&I" "oth-class-taut:3:a" "universal-cor")
6535qed
6536
6537AOT_theorem "ord=Eequiv:2": ‹x =E y → y =E x›
6538proof(rule CP)
6539  AOT_assume 1: ‹x =E y›
6540  AOT_hence 2: ‹x = y› by (metis "=E-simple:2" "vdash-properties:10") 
6541  AOT_have ‹O!x› using 1 by (meson "&E"(1) "=E-simple:1" "≡E"(1))
6542  AOT_hence ‹x =E x› using "ord=Eequiv:1" "→E" by blast
6543  AOT_thus ‹y =E x› using "rule=E"[rotated, OF 2] by fast
6544qed
6545
6546AOT_theorem "ord=Eequiv:3": ‹(x =E y & y =E z) → x =E z›
6547proof (rule CP)
6548  AOT_assume 1: ‹x =E y & y =E z›
6549  AOT_hence ‹x = y & y = z›
6550    by (metis "&I" "&E"(1) "&E"(2) "=E-simple:2" "vdash-properties:6")
6551  AOT_hence ‹x = z› by (metis "id-eq:3" "vdash-properties:6")
6552  moreover AOT_have ‹x =E x›
6553    using 1[THEN "&E"(1)] "&E"(1) "=E-simple:1" "≡E"(1) "ord=Eequiv:1" "→E" by blast
6554  ultimately AOT_show ‹x =E z›
6555    using "rule=E" by fast
6556qed
6557
6558AOT_theorem "ord-=E=:1": ‹(O!x ∨ O!y) → □(x = y ≡ x =E y)›
6559proof(rule CP)
6560  AOT_assume ‹O!x ∨ O!y›
6561  moreover {
6562    AOT_assume ‹O!x›
6563    AOT_hence ‹□O!x› by (metis "oa-facts:1" "vdash-properties:10")
6564    moreover {
6565      AOT_modally_strict {
6566        AOT_have ‹O!x → (x = y ≡ x =E y)›
6567        proof (rule "→I"; rule "≡I"; rule "→I")
6568          AOT_assume ‹O!x›
6569          AOT_hence ‹x =E x› by (metis "ord=Eequiv:1" "→E")
6570          moreover AOT_assume ‹x = y›
6571          ultimately AOT_show ‹x =E y› using "rule=E" by fast
6572        next
6573          AOT_assume ‹x =E y›
6574          AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
6575        qed
6576      }
6577      AOT_hence ‹□O!x → □(x = y ≡ x =E y)› by (metis "RM:1")
6578    }
6579    ultimately AOT_have ‹□(x = y ≡ x =E y)› using "→E" by blast
6580  }
6581  moreover {
6582    AOT_assume ‹O!y›
6583    AOT_hence ‹□O!y› by (metis "oa-facts:1" "vdash-properties:10")
6584    moreover {
6585      AOT_modally_strict {
6586        AOT_have ‹O!y → (x = y ≡ x =E y)›
6587        proof (rule "→I"; rule "≡I"; rule "→I")
6588          AOT_assume ‹O!y›
6589          AOT_hence ‹y =E y› by (metis "ord=Eequiv:1" "→E")
6590          moreover AOT_assume ‹x = y›
6591          ultimately AOT_show ‹x =E y› using "rule=E" id_sym by fast
6592        next
6593          AOT_assume ‹x =E y›
6594          AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
6595        qed
6596      }
6597      AOT_hence ‹□O!y → □(x = y ≡ x =E y)› by (metis "RM:1")
6598    }
6599    ultimately AOT_have ‹□(x = y ≡ x =E y)› using "→E" by blast
6600  }
6601  ultimately AOT_show ‹□(x = y ≡ x =E y)› by (metis "∨E"(3) "raa-cor:1")
6602qed
6603
6604AOT_theorem "ord-=E=:2": ‹O!y → [λx x = y]↓›
6605proof (rule "→I"; rule "safe-ext"[axiom_inst, THEN "→E"]; rule "&I")
6606  AOT_show ‹[λx x =E y]↓› by "cqt:2[lambda]"
6607next
6608  AOT_assume ‹O!y›
6609  AOT_hence 1: ‹□(x = y ≡ x =E y)› for x using "ord-=E=:1" "→E" "∨I" by blast
6610  AOT_have ‹□(x =E y ≡ x = y)› for x
6611    by (AOT_subst ‹x =E y ≡ x = y› ‹x = y ≡ x =E y›)
6612       (auto simp add: "Commutativity of ≡" 1)
6613  AOT_hence ‹∀x □(x =E y ≡ x = y)› by (rule GEN)
6614  AOT_thus ‹□∀x (x =E y ≡ x = y)› by (rule BF[THEN "→E"])
6615qed
6616
6617
6618AOT_theorem "ord-=E=:3": ‹[λxy O!x & O!y & x = y]↓›
6619proof (rule "safe-ext[2]"[axiom_inst, THEN "→E"]; rule "&I")
6620  AOT_show ‹[λxy O!x & O!y & x =E y]↓› by "cqt:2[lambda]"
6621next
6622  AOT_show ‹□∀x∀y ([O!]x & [O!]y & x =E y ≡ [O!]x & [O!]y & x = y)›
6623  proof (rule RN; rule GEN; rule GEN; rule "≡I"; rule "→I")
6624    AOT_modally_strict {
6625      AOT_show ‹[O!]x & [O!]y & x = y› if ‹[O!]x & [O!]y & x =E y› for x y
6626        by (metis "&I" "&E"(1) "Conjunction Simplification"(2) "=E-simple:2"
6627                  "modus-tollens:1" "raa-cor:1" that)
6628    }
6629  next
6630    AOT_modally_strict {
6631      AOT_show ‹[O!]x & [O!]y & x =E y› if ‹[O!]x & [O!]y & x = y› for x y
6632        apply(safe intro!: "&I")
6633          apply (metis that[THEN "&E"(1), THEN "&E"(1)])
6634         apply (metis that[THEN "&E"(1), THEN "&E"(2)])
6635        using "rule=E"[rotated, OF that[THEN "&E"(2)]]
6636              "ord=Eequiv:1"[THEN "→E", OF that[THEN "&E"(1), THEN "&E"(1)]] by fast
6637    }
6638  qed
6639qed
6640
6641AOT_theorem "ind-nec": ‹∀F ([F]x ≡ [F]y) → □∀F ([F]x ≡ [F]y)›
6642proof(rule "→I")
6643  AOT_assume ‹∀F ([F]x ≡ [F]y)›
6644  moreover AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]↓› by "cqt:2[lambda]"
6645  ultimately AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]x ≡ [λx □∀F ([F]x ≡ [F]y)]y›
6646    using "∀E" by blast
6647  moreover AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]y›
6648    apply (rule "β←C"(1))
6649      apply "cqt:2[lambda]"
6650     apply (fact "cqt:2[const_var]"[axiom_inst])
6651    by (simp add: RN GEN "oth-class-taut:3:a")
6652  ultimately AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]x› using "≡E" by blast
6653  AOT_thus ‹□∀F ([F]x ≡ [F]y)›
6654    using "β→C"(1) by blast
6655qed
6656
6657AOT_theorem "ord=E:1": ‹(O!x & O!y) → (∀F ([F]x ≡ [F]y) → x =E y)›
6658proof (rule "→I"; rule "→I")
6659  AOT_assume ‹∀F ([F]x ≡ [F]y)›
6660  AOT_hence ‹□∀F ([F]x ≡ [F]y)›
6661    using "ind-nec"[THEN "→E"] by blast
6662  moreover AOT_assume ‹O!x & O!y›
6663  ultimately AOT_have ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›
6664    using "&I" by blast
6665  AOT_thus ‹x =E y› using "=E-simple:1"[THEN "≡E"(2)] by blast
6666qed
6667
6668AOT_theorem "ord=E:2": ‹(O!x & O!y) → (∀F ([F]x ≡ [F]y) → x = y)›
6669proof (rule "→I"; rule "→I")
6670  AOT_assume ‹O!x & O!y›
6671  moreover AOT_assume ‹∀F ([F]x ≡ [F]y)›
6672  ultimately AOT_have ‹x =E y›
6673    using "ord=E:1" "→E" by blast
6674  AOT_thus ‹x = y› using "=E-simple:2"[THEN "→E"] by blast
6675qed
6676
6677AOT_theorem "ord=E2:1": ‹(O!x & O!y) → (x ≠ y ≡ [λz z =E x] ≠ [λz z =E y])›
6678proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6679  AOT_assume 0: ‹O!x & O!y›
6680  AOT_assume ‹x ≠ y›
6681  AOT_hence 1: ‹¬(x = y)› using "≡dfE"[OF "=-infix"] by blast
6682  AOT_assume ‹[λz z =E x] = [λz z =E y]›
6683  moreover AOT_have ‹[λz z =E x]x›
6684    apply (rule "β←C"(1))
6685      apply "cqt:2[lambda]"
6686     apply (fact "cqt:2[const_var]"[axiom_inst])
6687    using "ord=Eequiv:1"[THEN "→E", OF 0[THEN "&E"(1)]].
6688  ultimately AOT_have ‹[λz z =E y]x› using "rule=E" by fast
6689  AOT_hence ‹x =E y› using "β→C"(1) by blast
6690  AOT_hence ‹x = y› by (metis "=E-simple:2" "vdash-properties:6")
6691  AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
6692next
6693  AOT_assume ‹[λz z =E x] ≠ [λz z =E y]›
6694  AOT_hence 0: ‹¬([λz z =E x] = [λz z =E y])› using "≡dfE"[OF "=-infix"] by blast
6695  AOT_have ‹[λz z =E x]↓› by "cqt:2[lambda]"
6696  AOT_hence ‹[λz z =E x] = [λz z =E x]›
6697    by (metis "rule=I:1")
6698  moreover AOT_assume ‹x = y›
6699  ultimately AOT_have ‹[λz z =E x] = [λz z =E y]›
6700    using "rule=E" by fast
6701  AOT_thus ‹[λz z =E x] = [λz z =E y] & ¬([λz z =E x] = [λz z =E y])›
6702    using 0 "&I" by blast
6703qed
6704
6705AOT_theorem "ord=E2:2": ‹(O!x & O!y) → (x ≠ y ≡ [λz z = x] ≠ [λz z = y])›
6706proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6707  AOT_assume 0: ‹O!x & O!y›
6708  AOT_assume ‹x ≠ y›
6709  AOT_hence 1: ‹¬(x = y)› using "≡dfE"[OF "=-infix"] by blast
6710  AOT_assume ‹[λz z = x] = [λz z = y]›
6711  moreover AOT_have ‹[λz z = x]x›
6712    apply (rule "β←C"(1))
6713    apply (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6714     apply (fact "cqt:2[const_var]"[axiom_inst])
6715    by (simp add: "id-eq:1")
6716  ultimately AOT_have ‹[λz z = y]x› using "rule=E" by fast
6717  AOT_hence ‹x = y› using "β→C"(1) by blast
6718  AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
6719next
6720  AOT_assume 0: ‹O!x & O!y›
6721  AOT_assume ‹[λz z = x] ≠ [λz z = y]›
6722  AOT_hence 1: ‹¬([λz z = x] = [λz z = y])› using "≡dfE"[OF "=-infix"] by blast
6723  AOT_have ‹[λz z = x]↓› by (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6724  AOT_hence ‹[λz z = x] = [λz z = x]›
6725    by (metis "rule=I:1")
6726  moreover AOT_assume ‹x = y›
6727  ultimately AOT_have ‹[λz z = x] = [λz z = y]›
6728    using "rule=E" by fast
6729  AOT_thus ‹[λz z = x] = [λz z = y] & ¬([λz z = x] = [λz z = y])›
6730    using 1 "&I" by blast
6731qed
6732
6733AOT_theorem ordnecfail: ‹O!x → □¬∃F x[F]›
6734  by (meson "RM:1" "deduction-theorem" nocoder "oa-facts:1" "vdash-properties:10" "vdash-properties:1[2]")
6735
6736AOT_theorem "ab-obey:1": ‹(A!x & A!y) → (∀F (x[F] ≡ y[F]) → x = y)›
6737proof (rule "→I"; rule "→I")
6738  AOT_assume 1: ‹A!x & A!y›
6739  AOT_assume ‹∀F (x[F] ≡ y[F])›
6740  AOT_hence ‹x[F] ≡ y[F]› for F using "∀E" by blast
6741  AOT_hence ‹□(x[F] ≡ y[F])› for F by (metis "en-eq:6[1]" "≡E"(1))
6742  AOT_hence ‹∀F □(x[F] ≡ y[F])› by (rule GEN)
6743  AOT_hence ‹□∀F (x[F] ≡ y[F])› by (rule BF[THEN "→E"])
6744  AOT_thus ‹x = y›
6745    using "≡dfI"[OF "identity:1", OF "∨I"(2)] 1 "&I" by blast
6746qed
6747
6748AOT_theorem "ab-obey:2": ‹(∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])) → x ≠ y›
6749proof (rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6750  AOT_assume 1: ‹x = y›
6751  AOT_assume ‹∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])›
6752  moreover {
6753    AOT_assume ‹∃F (x[F] & ¬y[F])›
6754    then AOT_obtain F where ‹x[F] & ¬y[F]› using "∃E"[rotated] by blast
6755    moreover AOT_have ‹y[F]› using calculation[THEN "&E"(1)] 1 "rule=E" by fast
6756    ultimately AOT_have ‹p & ¬p› for p by (metis "Conjunction Simplification"(2) "modus-tollens:2" "raa-cor:3")
6757  }
6758  moreover {
6759    AOT_assume ‹∃F (y[F] & ¬x[F])›
6760    then AOT_obtain F where ‹y[F] & ¬x[F]› using "∃E"[rotated] by blast
6761    moreover AOT_have ‹¬y[F]› using calculation[THEN "&E"(2)] 1 "rule=E" by fast
6762    ultimately AOT_have ‹p & ¬p› for p by (metis "Conjunction Simplification"(1) "modus-tollens:1" "raa-cor:3")
6763  }
6764  ultimately AOT_show ‹p & ¬p› for p by (metis "∨E"(3) "raa-cor:1")
6765qed
6766
6767AOT_theorem "encoders-are-abstract": ‹∃F x[F] → A!x›
6768  by (meson "deduction-theorem" "≡E"(2) "modus-tollens:2" nocoder
6769            "oa-contingent:3" "vdash-properties:1[2]")
6770
6771AOT_theorem "denote=:1": ‹∀H∃x x[H]›
6772  by (rule GEN; rule "existence:2[1]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6773
6774AOT_theorem "denote=:2": ‹∀G∃x1...∃xn x1...xn[H]›
6775  by (rule GEN; rule "existence:2"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6776
6777AOT_theorem "denote=:2[2]": ‹∀G∃x1∃x2 x1x2[H]›
6778  by (rule GEN; rule "existence:2[2]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6779
6780AOT_theorem "denote=:2[3]": ‹∀G∃x1∃x2∃x3 x1x2x3[H]›
6781  by (rule GEN; rule "existence:2[3]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6782
6783AOT_theorem "denote=:2[4]": ‹∀G∃x1∃x2∃x3∃x4 x1x2x3x4[H]›
6784  by (rule GEN; rule "existence:2[4]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6785
6786AOT_theorem "denote=:3": ‹∃x x[Π] ≡ ∃H (H = Π)›
6787  using "existence:2[1]" "free-thms:1" "≡E"(2) "≡E"(5) "Commutativity of ≡" "≡Df" by blast
6788
6789AOT_theorem "denote=:4": ‹(∃x1...∃xn x1...xn[Π]) ≡ ∃H (H = Π)›
6790  using "existence:2" "free-thms:1" "≡E"(6) "≡Df" by blast
6791
6792AOT_theorem "denote=:4[2]": ‹(∃x1∃x2 x1x2[Π]) ≡ ∃H (H = Π)›
6793  using "existence:2[2]" "free-thms:1" "≡E"(6) "≡Df" by blast
6794
6795AOT_theorem "denote=:4[3]": ‹(∃x1∃x2∃x3 x1x2x3[Π]) ≡ ∃H (H = Π)›
6796  using "existence:2[3]" "free-thms:1" "≡E"(6) "≡Df" by blast
6797
6798AOT_theorem "denote=:4[4]": ‹(∃x1∃x2∃x3∃x4 x1x2x3x4[Π]) ≡ ∃H (H = Π)›
6799  using "existence:2[4]" "free-thms:1" "≡E"(6) "≡Df" by blast
6800
6801AOT_theorem "A-objects!": ‹∃!x (A!x & ∀F (x[F] ≡ φ{F}))›
6802proof (rule "uniqueness:1"[THEN "≡dfI"])
6803  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ φ{F})›
6804    using "A-objects"[axiom_inst] "∃E"[rotated] by blast
6805  AOT_have ‹(A!β & ∀F (β[F] ≡ φ{F})) → β = a› for β
6806  proof (rule "→I")
6807    AOT_assume β_prop: ‹[A!]β & ∀F (β[F] ≡ φ{F})›
6808    AOT_hence ‹β[F] ≡ φ{F}› for F using "∀E" "&E" by blast
6809    AOT_hence ‹β[F] ≡ a[F]› for F
6810      using a_prop[THEN "&E"(2)] "∀E" "≡E"(2) "≡E"(5) "Commutativity of ≡" by fast
6811    AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
6812    AOT_thus ‹β = a›
6813      using "ab-obey:1"[THEN "→E", OF "&I"[OF β_prop[THEN "&E"(1)], OF a_prop[THEN "&E"(1)]], THEN "→E"] by blast
6814  qed
6815  AOT_hence ‹∀β ((A!β & ∀F (β[F] ≡ φ{F})) → β = a)› by (rule GEN)
6816  AOT_thus ‹∃α ([A!]α & ∀F (α[F] ≡ φ{F}) & ∀β ([A!]β & ∀F (β[F] ≡ φ{F}) → β = α)) ›
6817    using "∃I" using a_prop "&I" by fast
6818qed
6819
6820AOT_theorem "obj-oth:1": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y))›
6821  using "A-objects!" by fast
6822
6823AOT_theorem "obj-oth:2": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y & [F]z))›
6824  using "A-objects!" by fast
6825
6826AOT_theorem "obj-oth:3": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y ∨ [F]z))›
6827  using "A-objects!" by fast
6828
6829AOT_theorem "obj-oth:4": ‹∃!x (A!x & ∀F (x[F] ≡ □[F]y))›
6830  using "A-objects!" by fast
6831
6832AOT_theorem "obj-oth:5": ‹∃!x (A!x & ∀F (x[F] ≡ F = G))›
6833  using "A-objects!" by fast
6834
6835AOT_theorem "obj-oth:6": ‹∃!x (A!x & ∀F (x[F] ≡ □∀y([G]y → [F]y)))›
6836  using "A-objects!" by fast
6837
6838AOT_theorem "A-descriptions": ‹ιx (A!x & ∀F (x[F] ≡ φ{F}))↓›
6839  by (rule "A-Exists:2"[THEN "≡E"(2)]; rule "RA[2]"; rule "A-objects!")
6840
6841AOT_act_theorem "thm-can-terms2": ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
6842  using "y-in:2" by blast
6843
6844AOT_theorem "can-ab2": ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F})) →  A!y›
6845proof(rule "→I")
6846  AOT_assume ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F}))›
6847  AOT_hence ‹𝒜(A!y & ∀F (y[F] ≡ φ{F}))›
6848    using "actual-desc:2"[THEN "→E"] by blast
6849  AOT_hence ‹𝒜A!y› by (metis "Act-Basic:2" "&E"(1) "≡E"(1))
6850  AOT_thus ‹A!y› by (metis "≡E"(2) "oa-facts:8")
6851qed
6852
6853AOT_act_theorem "desc-encode:1": ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
6854proof -
6855  AOT_have ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
6856    by (simp add: "A-descriptions")
6857  AOT_hence ‹A!ιx(A!x & ∀F (x[F] ≡ φ{F})) & ∀F (ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6858    using "y-in:3"[THEN "→E"] by blast
6859  AOT_thus ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
6860    using "&E" "∀E" by blast
6861qed
6862
6863AOT_act_theorem "desc-encode:2": ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G}›
6864  using "desc-encode:1".
6865
6866AOT_theorem "desc-nec-encode:1": ‹ιx (A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ 𝒜φ{F}›
6867proof -
6868  AOT_have 0: ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
6869    by (simp add: "A-descriptions")
6870  AOT_hence ‹𝒜(A!ιx(A!x & ∀F (x[F] ≡ φ{F})) & ∀F (ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}))›
6871    using "actual-desc:4"[THEN "→E"] by blast
6872  AOT_hence ‹𝒜∀F (ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6873    using "Act-Basic:2" "&E"(2) "≡E"(1) by blast
6874  AOT_hence ‹∀F 𝒜(ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6875    using "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]" by blast
6876  AOT_hence ‹𝒜(ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6877    using "∀E" by blast
6878  AOT_hence ‹𝒜ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ 𝒜φ{F}›
6879    using "Act-Basic:5" "≡E"(1) by blast
6880  AOT_thus ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ 𝒜φ{F}›
6881    using "en-eq:10[1]"[unvarify x1, OF 0] "≡E"(6) by blast
6882qed
6883
6884AOT_theorem "desc-nec-encode:2": ‹ιx (A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ 𝒜φ{G}›
6885  using "desc-nec-encode:1".
6886
6887AOT_theorem "Box-desc-encode:1": ‹□φ{G} → ιx(A!x & ∀F (x[F] ≡ φ{G}))[G]›
6888  by (rule "→I"; rule "desc-nec-encode:2"[THEN "≡E"(2)])
6889     (meson "nec-imp-act" "vdash-properties:10")
6890
6891AOT_theorem "Box-desc-encode:2": ‹□φ{G} → □(ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
6892proof(rule CP)
6893  AOT_assume ‹□φ{G}›
6894  AOT_hence ‹□□φ{G}› by (metis "S5Basic:6" "≡E"(1))
6895  moreover AOT_have ‹□□φ{G} → □(ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
6896  proof (rule RM; rule "→I")
6897    AOT_modally_strict {
6898      AOT_assume 1: ‹□φ{G}›
6899      AOT_hence ‹ιx(A!x & ∀F (x[F] ≡ φ{G}))[G]› using "Box-desc-encode:1" "→E" by blast
6900      moreover AOT_have ‹φ{G}› using 1 by (meson "qml:2" "vdash-properties:10" "vdash-properties:1[2]")
6901      ultimately AOT_show ‹ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G}›
6902        using "deduction-theorem" "≡I" by simp
6903    }
6904  qed
6905  ultimately AOT_show ‹□(ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})› using "→E" by blast
6906qed
6907
6908definition rigid_condition where ‹rigid_condition φ ≡ ∀v . [v ⊨ ∀α (φ{α} → □φ{α})]›
6909syntax rigid_condition :: ‹id_position ⇒ AOT_prop› ("RIGID'_CONDITION'(_')")
6910
6911AOT_theorem "strict-can:1[E]": assumes ‹RIGID_CONDITION(φ)›
6912  shows ‹∀α (φ{α} → □φ{α})›
6913  using assms[unfolded rigid_condition_def] by auto
6914
6915AOT_theorem "strict-can:1[I]":
6916  assumes ‹⊢□ ∀α (φ{α} → □φ{α})›
6917  shows ‹RIGID_CONDITION(φ)›
6918  using assms rigid_condition_def by auto
6919
6920AOT_theorem "box-phi-a:1": assumes ‹RIGID_CONDITION(φ)›
6921  shows ‹(A!x  & ∀F (x[F] ≡ φ{F})) → □(A!x & ∀F (x[F] ≡ φ{F}))›
6922proof (rule "→I")
6923  AOT_assume a: ‹A!x & ∀F (x[F] ≡ φ{F})›
6924  AOT_hence b: ‹□A!x› by (metis "Conjunction Simplification"(1) "oa-facts:2" "vdash-properties:10")
6925  AOT_have ‹x[F] ≡ φ{F}› for F using a[THEN "&E"(2)] "∀E" by blast
6926  moreover AOT_have ‹□(x[F] → □x[F])› for F by (meson "pre-en-eq:1[1]" RN)
6927  moreover AOT_have ‹□(φ{F} → □φ{F})› for F using RN "strict-can:1[E]"[OF assms] "∀E" by blast
6928  ultimately AOT_have ‹□(x[F] ≡ φ{F})› for F
6929    using "sc-eq-box-box:5" "qml:2"[axiom_inst, THEN "→E"] "→E" "&I" by metis
6930  AOT_hence ‹∀F □(x[F] ≡ φ{F})› by (rule GEN)
6931  AOT_hence ‹□∀F (x[F] ≡ φ{F})› by (rule BF[THEN "→E"])
6932  AOT_thus ‹□([A!]x & ∀F (x[F] ≡ φ{F}))›
6933    using b "KBasic:3" "≡S"(1) "≡E"(2) by blast
6934qed
6935
6936AOT_theorem "box-phi-a:2": assumes ‹RIGID_CONDITION(φ)›
6937  shows ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
6938proof(rule "→I")
6939  AOT_assume ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F}))›
6940  AOT_hence ‹𝒜(A!y & ∀F (y[F] ≡ φ{F}))› using "actual-desc:2"[THEN "→E"] by fast
6941  AOT_hence abs: ‹𝒜A!y› and ‹𝒜∀F (y[F] ≡ φ{F})›
6942    using "Act-Basic:2" "&E" "≡E"(1) by blast+
6943  AOT_hence ‹∀F 𝒜(y[F] ≡ φ{F})› by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
6944  AOT_hence ‹𝒜(y[F] ≡ φ{F})› for F using "∀E" by blast
6945  AOT_hence ‹𝒜y[F] ≡ 𝒜φ{F}› for F by (metis "Act-Basic:5" "≡E"(1)) 
6946  AOT_hence ‹y[F] ≡ φ{F}› for F
6947    using "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=F], THEN RN]]
6948    by (metis "en-eq:10[1]" "≡E"(6))
6949  AOT_hence ‹∀F (y[F] ≡ φ{F})› by (rule GEN)
6950  AOT_thus ‹[A!]y & ∀F (y[F] ≡ φ{F})› using abs "&I" "≡E"(2) "oa-facts:8" by blast
6951qed
6952
6953AOT_theorem "box-phi-a:3": assumes ‹RIGID_CONDITION(φ)›
6954  shows ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
6955  using "desc-nec-encode:2"
6956    "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=F], THEN RN]]
6957    "≡E"(5) by blast
6958
6959AOT_define Null :: ‹τ ⇒ φ› ("Null'(_')") 
6960  "df-null-uni:1": ‹Null(x) ≡df A!x & ¬∃F x[F]›
6961
6962AOT_define Universal :: ‹τ ⇒ φ› ("Universal'(_')")
6963  "df-null-uni:2": ‹Universal(x) ≡df A!x & ∀F x[F]›
6964
6965AOT_theorem "null-uni-uniq:1": ‹∃!x Null(x)›
6966proof (rule "uniqueness:1"[THEN "≡dfI"])
6967  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ¬(F = F))›
6968    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
6969  AOT_have a_null: ‹¬a[F]› for F
6970  proof (rule "raa-cor:2")
6971    AOT_assume ‹a[F]›
6972    AOT_hence ‹¬(F = F)› using a_prop[THEN "&E"(2)] "∀E" "≡E" by blast
6973    AOT_hence ‹F = F & ¬(F = F)› by (metis "id-eq:1" "raa-cor:3")
6974    AOT_thus ‹p & ¬p› for p  by (metis "raa-cor:1")
6975  qed
6976  AOT_have ‹Null(a) & ∀β (Null(β) → β = a)›
6977  proof (rule "&I")
6978    AOT_have ‹¬∃F a[F]› using a_null by (metis "instantiation" "reductio-aa:1")
6979    AOT_thus ‹Null(a)›
6980      using "df-null-uni:1"[THEN "≡dfI"] a_prop[THEN "&E"(1)] "&I" by metis
6981  next
6982    AOT_show ‹∀β (Null(β) → β = a)›
6983    proof (rule GEN; rule "→I")
6984      fix β
6985      AOT_assume a: ‹Null(β)›
6986      AOT_hence ‹¬∃F β[F]›
6987        using "df-null-uni:1"[THEN "≡dfE"] "&E" by blast
6988      AOT_hence β_null: ‹¬β[F]› for F by (metis "existential:2[const_var]" "reductio-aa:1")
6989      AOT_have ‹∀F (β[F] ≡ a[F])›
6990        apply (rule GEN; rule "≡I"; rule CP)
6991        using "raa-cor:3" β_null a_null by blast+
6992      moreover AOT_have ‹A!β› using a "df-null-uni:1"[THEN "≡dfE"] "&E" by blast
6993      ultimately AOT_show ‹β = a›
6994        using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" by blast
6995    qed
6996  qed
6997  AOT_thus ‹∃α (Null(α) & ∀β (Null(β) → β = α))› using "∃I"(2) by fast
6998qed
6999
7000AOT_theorem "null-uni-uniq:2": ‹∃!x Universal(x)›
7001proof (rule "uniqueness:1"[THEN "≡dfI"])
7002  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ F = F)›
7003    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7004  AOT_hence aF: ‹a[F]› for F using "&E" "∀E" "≡E" "id-eq:1" by fast
7005  AOT_hence ‹Universal(a)›
7006    using "df-null-uni:2"[THEN "≡dfI"] "&I" a_prop[THEN "&E"(1)] GEN by blast
7007  moreover AOT_have ‹∀β (Universal(β) → β = a)›
7008  proof (rule GEN; rule "→I")
7009    fix β
7010    AOT_assume ‹Universal(β)›
7011    AOT_hence abs_β: ‹A!β› and ‹β[F]› for F using "df-null-uni:2"[THEN "≡dfE"] "&E" "∀E" by blast+
7012    AOT_hence ‹β[F] ≡ a[F]› for F using aF by (metis "deduction-theorem" "≡I")
7013    AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
7014    AOT_thus ‹β = a›
7015      using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" abs_β by blast
7016  qed
7017  ultimately AOT_show ‹∃α (Universal(α) & ∀β (Universal(β) → β = α))›
7018    using "&I" "∃I" by fast
7019qed
7020
7021AOT_theorem "null-uni-uniq:3": ‹ιx Null(x)↓›
7022  using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:1" by blast
7023
7024AOT_theorem "null-uni-uniq:4": ‹ιx Universal(x)↓›
7025  using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:2" by blast
7026
7027AOT_define Null_object :: ‹κs› (‹a∅›)
7028  "df-null-uni-terms:1": ‹a∅ =df ιx Null(x)›
7029
7030AOT_define Universal_object :: ‹κs› (‹aV›)
7031  "df-null-uni-terms:2": ‹aV =df ιx Universal(x)›
7032
7033AOT_theorem "null-uni-facts:1": ‹Null(x) → □Null(x)›
7034proof (rule "→I")
7035  AOT_assume ‹Null(x)›
7036  AOT_hence x_abs: ‹A!x› and x_null: ‹¬∃F x[F]›
7037    using "df-null-uni:1"[THEN "≡dfE"] "&E" by blast+
7038  AOT_have ‹¬x[F]› for F using x_null
7039    using "existential:2[const_var]" "reductio-aa:1"
7040    by metis
7041  AOT_hence ‹□¬x[F]› for F by (metis "en-eq:7[1]" "≡E"(1))
7042  AOT_hence ‹∀F □¬x[F]› by (rule GEN)
7043  AOT_hence ‹□∀F ¬x[F]› by (rule BF[THEN "→E"])
7044  moreover AOT_have ‹□∀F ¬x[F] → □¬∃F x[F]›
7045    apply (rule RM)
7046    by (metis (full_types) "instantiation" "cqt:2[const_var]" "deduction-theorem"
7047                           "reductio-aa:1" "rule-ui:1" "vdash-properties:1[2]")
7048  ultimately AOT_have ‹□¬∃F x[F]›
7049    by (metis "→E")
7050  moreover AOT_have ‹□A!x› using x_abs
7051    using "oa-facts:2" "vdash-properties:10" by blast
7052  ultimately AOT_have r: ‹□(A!x & ¬∃F x[F])›
7053    by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
7054  AOT_show ‹□Null(x)›
7055    by (AOT_subst ‹Null(x)› ‹A!x & ¬∃F x[F]›)
7056       (auto simp: "df-null-uni:1" "≡Df" r)
7057qed  
7058
7059AOT_theorem "null-uni-facts:2": ‹Universal(x) → □Universal(x)›
7060proof (rule "→I")
7061  AOT_assume ‹Universal(x)›
7062  AOT_hence x_abs: ‹A!x› and x_univ: ‹∀F x[F]›
7063    using "df-null-uni:2"[THEN "≡dfE"] "&E" by blast+
7064  AOT_have ‹x[F]› for F using x_univ "∀E" by blast
7065  AOT_hence ‹□x[F]› for F by (metis "en-eq:2[1]" "≡E"(1))
7066  AOT_hence ‹∀F □x[F]› by (rule GEN)
7067  AOT_hence ‹□∀F x[F]› by (rule BF[THEN "→E"])
7068  moreover AOT_have ‹□A!x› using x_abs
7069    using "oa-facts:2" "vdash-properties:10" by blast
7070  ultimately AOT_have r: ‹□(A!x & ∀F x[F])›
7071    by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
7072  AOT_show ‹□Universal(x)›
7073    by (AOT_subst ‹Universal(x)› ‹A!x & ∀F x[F]›)
7074       (auto simp add: "df-null-uni:2" "≡Df" r)
7075qed
7076
7077AOT_theorem "null-uni-facts:3": ‹Null(a∅)›
7078  apply (rule "=dfI"(2)[OF "df-null-uni-terms:1"])
7079   apply (simp add: "null-uni-uniq:3")
7080  using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:3"]
7081    "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:1"[unvarify x, THEN RN, OF "null-uni-uniq:3"], THEN "≡E"(1)]
7082  by blast
7083
7084AOT_theorem "null-uni-facts:4": ‹Universal(aV)›
7085  apply (rule "=dfI"(2)[OF "df-null-uni-terms:2"])
7086   apply (simp add: "null-uni-uniq:4")
7087  using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:4"]
7088    "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:2"[unvarify x, THEN RN, OF "null-uni-uniq:4"], THEN "≡E"(1)]
7089  by blast
7090
7091AOT_theorem "null-uni-facts:5": ‹a∅ ≠ aV›
7092proof (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"];
7093    rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"];
7094    rule "≡dfI"[OF "=-infix"];
7095    rule "raa-cor:2")
7096  AOT_obtain x where nullx: ‹Null(x)›
7097    by (metis "instantiation" "df-null-uni-terms:1" "existential:1" "null-uni-facts:3"
7098              "null-uni-uniq:3" "rule-id-df:2:b[zero]")
7099  AOT_hence act_null: ‹𝒜Null(x)› by (metis "nec-imp-act" "null-uni-facts:1" "vdash-properties:10")
7100  AOT_assume ‹ιx Null(x) = ιx Universal(x)›
7101  AOT_hence ‹𝒜∀x(Null(x) ≡ Universal(x))›
7102    using "actual-desc:5"[THEN "→E"] by blast
7103  AOT_hence ‹∀x 𝒜(Null(x) ≡ Universal(x))›
7104    by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
7105  AOT_hence ‹𝒜Null(x) ≡ 𝒜Universal(x)›
7106    using "Act-Basic:5" "≡E"(1) "rule-ui:3" by blast
7107  AOT_hence ‹𝒜Universal(x)› using act_null "≡E" by blast
7108  AOT_hence ‹Universal(x)› by (metis RN "≡E"(1) "null-uni-facts:2" "sc-eq-fur:2" "vdash-properties:10")
7109  AOT_hence ‹∀F x[F]› using "≡dfE"[OF "df-null-uni:2"] "&E" by metis
7110  moreover AOT_have ‹¬∃F x[F]› using nullx "≡dfE"[OF "df-null-uni:1"] "&E" by metis
7111  ultimately AOT_show ‹p & ¬p› for p by (metis "cqt-further:1" "raa-cor:3" "vdash-properties:10")
7112qed
7113
7114AOT_theorem "null-uni-facts:6": ‹a∅ = ιx(A!x & ∀F (x[F] ≡ F ≠ F))›
7115proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7116  AOT_show ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F))↓›
7117    by (simp add: "A-descriptions")
7118next
7119  AOT_show ‹a∅↓›
7120    by (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7121       (simp add: "null-uni-uniq:3")
7122next
7123  AOT_have ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F))↓›
7124    by (simp add: "A-descriptions")
7125  AOT_hence 1: ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F)) = ιx([A!]x & ∀F (x[F] ≡ F ≠ F))›
7126    using "rule=I:1" by blast
7127  AOT_show ‹[A!]a∅ & [A!]ιx([A!]x & ∀F (x[F] ≡ F ≠ F))›
7128    apply (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"]; rule "&I")
7129    apply (meson "≡dfE" "Conjunction Simplification"(1) "df-null-uni:1" "df-null-uni-terms:1" "null-uni-facts:3" "null-uni-uniq:3" "rule-id-df:2:a[zero]" "vdash-properties:10")
7130    using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7131next
7132  AOT_show ‹∀F (a∅[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F])›
7133  proof (rule GEN)
7134    fix F
7135    AOT_have ‹¬a∅[F]›
7136      by (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7137         (metis (no_types, lifting) "≡dfE" "&E"(2) "∨I"(2) "∨E"(3)
7138                "df-null-uni:1" "df-null-uni-terms:1" "existential:2[const_var]" "null-uni-facts:3"
7139                "raa-cor:2" "rule-id-df:2:a[zero]" "russell-axiom[enc,1].ψ_denotes_asm")
7140    moreover AOT_have ‹¬ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7141    proof(rule "raa-cor:2")
7142      AOT_assume 0: ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7143      AOT_hence ‹𝒜(F ≠ F)› using "desc-nec-encode:2"[THEN "≡E"(1), OF 0] by blast
7144      moreover AOT_have ‹¬𝒜(F ≠ F)›
7145        using "≡dfE" "id-act:2" "id-eq:1" "≡E"(2) "=-infix" "raa-cor:3" by blast
7146      ultimately AOT_show ‹𝒜(F ≠ F) & ¬𝒜(F ≠ F)› by (rule "&I")
7147    qed
7148    ultimately AOT_show ‹a∅[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7149      using "deduction-theorem" "≡I" "raa-cor:4" by blast
7150  qed
7151qed
7152
7153AOT_theorem "null-uni-facts:7": ‹aV = ιx(A!x & ∀F (x[F] ≡ F = F))›
7154proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7155  AOT_show ‹ιx([A!]x & ∀F (x[F] ≡ F = F))↓›
7156    by (simp add: "A-descriptions")
7157next
7158  AOT_show ‹aV↓›
7159    by (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7160       (simp add: "null-uni-uniq:4")
7161next
7162  AOT_have ‹ιx([A!]x & ∀F (x[F] ≡ F = F))↓›
7163    by (simp add: "A-descriptions")
7164  AOT_hence 1: ‹ιx([A!]x & ∀F (x[F] ≡ F = F)) = ιx([A!]x & ∀F (x[F] ≡ F = F))›
7165    using "rule=I:1" by blast
7166  AOT_show ‹[A!]aV & [A!]ιx([A!]x & ∀F (x[F] ≡ F = F))›
7167    apply (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"]; rule "&I")
7168    apply (meson "≡dfE" "Conjunction Simplification"(1) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-df:2:a[zero]" "vdash-properties:10")
7169    using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7170next
7171  AOT_show ‹∀F (aV[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F = F))[F])›
7172  proof (rule GEN)
7173    fix F
7174    AOT_have ‹aV[F]›
7175      apply (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7176      using "≡dfE" "&E"(2) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-df:2:a[zero]" "rule-ui:3" by blast
7177    moreover AOT_have ‹ιx([A!]x & ∀F (x[F] ≡ F = F))[F]›
7178      using "RA[2]" "desc-nec-encode:2" "id-eq:1" "≡E"(2) by fastforce
7179    ultimately AOT_show ‹aV[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F = F))[F]›
7180      using "deduction-theorem" "≡I" by simp
7181  qed
7182qed
7183
7184AOT_theorem "aclassical:1": ‹∀R∃x∃y(A!x & A!y & x ≠ y & [λz [R]zx] = [λz [R]zy])›
7185proof(rule GEN)
7186  fix R
7187  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]zy] & ¬y[F]))›
7188    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7189  AOT_have a_enc: ‹a[λz [R]za]›
7190  proof (rule "raa-cor:1")
7191    AOT_assume 0: ‹¬a[λz [R]za]›
7192    AOT_hence ‹¬∃y(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7193      by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«[λz [R]za]»"],
7194                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7195         "cqt:2[lambda]"
7196    AOT_hence ‹∀y ¬(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7197      using "cqt-further:4" "vdash-properties:10" by blast
7198    AOT_hence ‹¬(A!a & [λz [R]za] = [λz [R]za] & ¬a[λz [R]za])› using "∀E" by blast
7199    AOT_hence ‹(A!a & [λz [R]za] = [λz [R]za]) → a[λz [R]za]›
7200      by (metis "&I" "deduction-theorem" "raa-cor:3")
7201    moreover AOT_have ‹[λz [R]za] = [λz [R]za]›
7202      by (rule "=I") "cqt:2[lambda]"
7203    ultimately AOT_have ‹a[λz [R]za]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7204    AOT_thus ‹a[λz [R]za] & ¬a[λz [R]za]›
7205      using 0 "&I" by blast
7206  qed
7207  AOT_hence ‹∃y(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7208    by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7209  then AOT_obtain b where b_prop: ‹A!b & [λz [R]za] = [λz [R]zb] & ¬b[λz [R]za]›
7210    using "∃E"[rotated] by blast
7211  AOT_have ‹a ≠ b›
7212    apply (rule "≡dfI"[OF "=-infix"])
7213    using a_enc b_prop[THEN "&E"(2)]
7214    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7215  AOT_hence ‹A!a & A!b & a ≠ b & [λz [R]za] = [λz [R]zb]›
7216    using b_prop "&E" a_prop "&I" by meson
7217  AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λz [R]za] = [λz [R]zy])› by (rule "∃I")
7218  AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λz [R]zx] = [λz [R]zy])› by (rule "∃I")
7219qed
7220
7221AOT_theorem "aclassical:2": ‹∀R∃x∃y(A!x & A!y & x ≠ y & [λz [R]xz] = [λz [R]yz])›
7222proof(rule GEN)
7223  fix R
7224  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]yz] & ¬y[F]))›
7225    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7226  AOT_have a_enc: ‹a[λz [R]az]›
7227  proof (rule "raa-cor:1")
7228    AOT_assume 0: ‹¬a[λz [R]az]›
7229    AOT_hence ‹¬∃y(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7230      by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«[λz [R]az]»"],
7231                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7232         "cqt:2[lambda]"
7233    AOT_hence ‹∀y ¬(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7234      using "cqt-further:4" "vdash-properties:10" by blast
7235    AOT_hence ‹¬(A!a & [λz [R]az] = [λz [R]az] & ¬a[λz [R]az])› using "∀E" by blast
7236    AOT_hence ‹(A!a & [λz [R]az] = [λz [R]az]) → a[λz [R]az]›
7237      by (metis "&I" "deduction-theorem" "raa-cor:3")
7238    moreover AOT_have ‹[λz [R]az] = [λz [R]az]›
7239      by (rule "=I") "cqt:2[lambda]"
7240    ultimately AOT_have ‹a[λz [R]az]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7241    AOT_thus ‹a[λz [R]az] & ¬a[λz [R]az]›
7242      using 0 "&I" by blast
7243  qed
7244  AOT_hence ‹∃y(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7245    by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7246  then AOT_obtain b where b_prop: ‹A!b & [λz [R]az] = [λz [R]bz] & ¬b[λz [R]az]›
7247    using "∃E"[rotated] by blast
7248  AOT_have ‹a ≠ b›
7249    apply (rule "≡dfI"[OF "=-infix"])
7250    using a_enc b_prop[THEN "&E"(2)]
7251    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7252  AOT_hence ‹A!a & A!b & a ≠ b & [λz [R]az] = [λz [R]bz]›
7253    using b_prop "&E" a_prop "&I" by meson
7254  AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λz [R]az] = [λz [R]yz])› by (rule "∃I")
7255  AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λz [R]xz] = [λz [R]yz])› by (rule "∃I")
7256qed
7257
7258AOT_theorem "aclassical:3": ‹∀F∃x∃y(A!x & A!y & x ≠ y & [λ [F]x] = [λ [F]y])›
7259proof(rule GEN)
7260  fix R
7261  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]y] & ¬y[F]))›
7262    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7263  AOT_have ‹[λz [R]a]↓› by "cqt:2[lambda]"
7264  (* TODO: S should no longer be necessary *)
7265  then AOT_obtain S where S_def: ‹S = [λz [R]a]›
7266    by (metis "instantiation" "rule=I:1" "existential:1" id_sym)
7267  AOT_have a_enc: ‹a[S]›
7268  proof (rule "raa-cor:1")
7269    AOT_assume 0: ‹¬a[S]›
7270    AOT_hence ‹¬∃y(A!y & S = [λz [R]y] & ¬y[S])›
7271      by (rule a_prop[THEN "&E"(2), THEN "∀E"(2)[where β=S],
7272                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated]) 
7273    AOT_hence ‹∀y ¬(A!y & S = [λz [R]y] & ¬y[S])›
7274      using "cqt-further:4" "vdash-properties:10" by blast
7275    AOT_hence ‹¬(A!a & S = [λz [R]a] & ¬a[S])› using "∀E" by blast
7276    AOT_hence ‹(A!a & S = [λz [R]a]) → a[S]›
7277      by (metis "&I" "deduction-theorem" "raa-cor:3")
7278    moreover AOT_have ‹S = [λz [R]a]› using S_def .
7279    ultimately AOT_have ‹a[S]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7280    AOT_thus ‹a[λz [R]a] & ¬a[λz [R]a]›  by (metis "0" "raa-cor:3") 
7281  qed
7282  AOT_hence ‹∃y(A!y & S = [λz [R]y] & ¬y[S])›
7283    by (rule a_prop[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1), rotated])
7284  then AOT_obtain b where b_prop: ‹A!b & S = [λz [R]b] & ¬b[S]›
7285    using "∃E"[rotated] by blast
7286  AOT_have 1: ‹a ≠ b›
7287    apply (rule "≡dfI"[OF "=-infix"])
7288    using a_enc b_prop[THEN "&E"(2)]
7289    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7290  AOT_have a: ‹[λ [R]a] = ([R]a)›
7291    apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7292    by (meson "log-prop-prop:2")
7293  AOT_have b: ‹[λ [R]b] = ([R]b)›
7294    apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7295    by (meson "log-prop-prop:2")
7296  AOT_have ‹[λ [R]a] = [λ [R]b]›
7297    apply (rule "rule=E"[rotated, OF a[THEN id_sym]])
7298    apply (rule "rule=E"[rotated, OF b[THEN id_sym]])
7299    apply (rule "identity:4"[THEN "≡dfI", OF "&I", rotated])
7300     apply (rule "rule=E"[rotated, OF S_def])
7301    using b_prop "&E" apply blast
7302    apply (safe intro!: "&I")
7303    by (simp add: "log-prop-prop:2")+
7304  AOT_hence ‹A!a & A!b & a ≠ b & [λ [R]a] = [λ [R]b]›
7305    using 1 a_prop[THEN "&E"(1)] b_prop[THEN "&E"(1), THEN "&E"(1)] "&I" by auto
7306  AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λ [R]a] = [λ [R]y])› by (rule "∃I")
7307  AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λ [R]x] = [λ [R]y])› by (rule "∃I")
7308qed
7309
7310AOT_theorem aclassical2: ‹∃x∃y (A!x & A!y & x ≠ y & ∀F ([F]x ≡ [F]y))›
7311proof -
7312  AOT_have ‹∃x ∃y ([A!]x & [A!]y & x ≠ y &
7313               [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7314    by (rule "aclassical:1"[THEN "∀E"(1)[where τ="«[λxy ∀F ([F]x ≡ [F]y)]»"]])
7315       "cqt:2[lambda]"
7316  then AOT_obtain x where ‹∃y ([A!]x & [A!]y & x ≠ y &
7317               [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7318    using "∃E"[rotated] by blast
7319  then AOT_obtain y where 0: ‹([A!]x & [A!]y & x ≠ y &
7320               [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7321    using "∃E"[rotated] by blast
7322  AOT_have ‹[λz [λxy ∀F ([F]x ≡ [F]y)]zx]x›
7323    apply (rule "β←C"(1))
7324      apply "cqt:2[lambda]"
7325     apply (fact "cqt:2[const_var]"[axiom_inst])
7326    apply (rule "β←C"(1))
7327      apply "cqt:2[lambda]"
7328    apply (simp add: "&I" "ex:1:a" prod_denotesI "rule-ui:3")
7329    by (simp add: "oth-class-taut:3:a" "universal-cor")
7330  AOT_hence ‹[λz [λxy ∀F ([F]x ≡ [F]y)]zy]x›
7331    by (rule "rule=E"[rotated, OF 0[THEN "&E"(2)]])
7332  AOT_hence ‹[λxy ∀F ([F]x ≡ [F]y)]xy›
7333    by (rule "β→C"(1))
7334  AOT_hence ‹∀F ([F]x ≡ [F]y)›
7335    using "β→C"(1) old.prod.case by fast
7336  AOT_hence ‹[A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y)› using 0 "&E" "&I" by blast
7337  AOT_hence ‹∃y ([A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y))› by (rule "∃I")
7338  AOT_thus ‹∃x∃y ([A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y))› by (rule "∃I"(2))
7339qed
7340
7341AOT_theorem "kirchner-thm:1": ‹[λx φ{x}]↓ ≡ □∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7342proof(rule "≡I"; rule "→I")
7343  AOT_assume ‹[λx φ{x}]↓›
7344  AOT_hence ‹□[λx φ{x}]↓› by (metis "exist-nec" "vdash-properties:10")
7345  moreover AOT_have ‹□[λx φ{x}]↓ → □∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7346  proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7347    AOT_modally_strict {
7348      fix x y
7349      AOT_assume 0: ‹[λx φ{x}]↓›
7350      moreover AOT_assume ‹∀F([F]x ≡ [F]y)›
7351      ultimately AOT_have ‹[λx φ{x}]x ≡ [λx φ{x}]y›
7352        using "∀E" by blast
7353      AOT_thus ‹(φ{x} ≡ φ{y})›
7354        using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7355    }
7356  qed
7357  ultimately AOT_show ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7358    using "→E" by blast
7359next
7360  AOT_have ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y})) → □∀y(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7361  proof(rule "RM:1"; rule "→I"; rule GEN)
7362    AOT_modally_strict {
7363      AOT_assume ‹∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7364      AOT_hence indisc: ‹φ{x} ≡ φ{y}› if ‹∀F([F]x ≡ [F]y)› for x y
7365        using "∀E"(2) "→E" that by blast
7366      AOT_show ‹(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})› for y
7367      proof (rule "raa-cor:1")
7368        AOT_assume ‹¬(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7369        AOT_hence ‹(∃x(∀F([F]x ≡ [F]y) & φ{x}) & ¬φ{y}) ∨ (¬(∃x(∀F([F]x ≡ [F]y) & φ{x})) & φ{y})›
7370          using "≡E"(1) "oth-class-taut:4:h" by blast
7371        moreover {
7372          AOT_assume 0: ‹∃x(∀F([F]x ≡ [F]y) & φ{x}) & ¬φ{y}›
7373          AOT_obtain a where ‹∀F([F]a ≡ [F]y) & φ{a}›
7374            using "∃E"[rotated, OF 0[THEN "&E"(1)]]  by blast
7375          AOT_hence ‹φ{y}› using indisc[THEN "≡E"(1)] "&E" by blast
7376          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7377        }
7378        moreover {
7379          AOT_assume 0: ‹(¬(∃x(∀F([F]x ≡ [F]y) & φ{x})) & φ{y})›
7380          AOT_hence ‹∀x ¬(∀F([F]x ≡ [F]y) & φ{x})›
7381            using "&E"(1) "cqt-further:4" "→E" by blast
7382          AOT_hence ‹¬(∀F([F]y ≡ [F]y) & φ{y})› using "∀E" by blast
7383          AOT_hence ‹¬∀F([F]y ≡ [F]y) ∨ ¬φ{y}›
7384            using "≡E"(1) "oth-class-taut:5:c" by blast
7385          moreover AOT_have ‹∀F([F]y ≡ [F]y)› by (simp add: "oth-class-taut:3:a" "universal-cor")
7386          ultimately AOT_have ‹¬φ{y}› by (metis "¬¬I" "∨E"(2))
7387          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7388        }
7389        ultimately AOT_show ‹p & ¬p› for p using "∨E"(3) "raa-cor:1" by blast
7390      qed
7391    }
7392  qed
7393  moreover AOT_assume ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7394  ultimately AOT_have ‹□∀y(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7395    using "→E" by blast
7396  AOT_thus ‹[λx φ{x}]↓›
7397    by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7398qed
7399
7400AOT_theorem "kirchner-thm:2": ‹[λx1...xn φ{x1...xn}]↓ ≡ □∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7401proof(rule "≡I"; rule "→I")
7402  AOT_assume ‹[λx1...xn φ{x1...xn}]↓›
7403  AOT_hence ‹□[λx1...xn φ{x1...xn}]↓› by (metis "exist-nec" "vdash-properties:10")
7404  moreover AOT_have ‹□[λx1...xn φ{x1...xn}]↓ → □∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7405  proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7406    AOT_modally_strict {
7407      fix x1xn y1yn :: ‹'a AOT_var›
7408      AOT_assume 0: ‹[λx1...xn φ{x1...xn}]↓›
7409      moreover AOT_assume ‹∀F([F]x1...xn ≡ [F]y1...yn)›
7410      ultimately AOT_have ‹[λx1...xn φ{x1...xn}]x1...xn ≡ [λx1...xn φ{x1...xn}]y1...yn›
7411        using "∀E" by blast
7412      AOT_thus ‹(φ{x1...xn} ≡ φ{y1...yn})›
7413        using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7414    }
7415  qed
7416  ultimately AOT_show ‹□∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7417    using "→E" by blast
7418next
7419  AOT_have ‹□(∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))) →
7420            □∀y1...∀yn((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn})›
7421  proof(rule "RM:1"; rule "→I"; rule GEN)
7422    AOT_modally_strict {
7423      AOT_assume ‹∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7424      AOT_hence indisc: ‹φ{x1...xn} ≡ φ{y1...yn}› if ‹∀F([F]x1...xn ≡ [F]y1...yn)› for x1xn y1yn
7425        using "∀E"(2) "→E" that by blast
7426      AOT_show ‹(∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn}› for y1yn
7427      proof (rule "raa-cor:1")
7428        AOT_assume ‹¬((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn})›
7429        AOT_hence ‹((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) & ¬φ{y1...yn}) ∨
7430                    (¬(∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) & φ{y1...yn})›
7431          using "≡E"(1) "oth-class-taut:4:h" by blast
7432        moreover {
7433          AOT_assume 0: ‹(∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) & ¬φ{y1...yn}›
7434          AOT_obtain a1an where ‹∀F([F]a1...an ≡ [F]y1...yn) & φ{a1...an}›
7435            using "∃E"[rotated, OF 0[THEN "&E"(1)]]  by blast
7436          AOT_hence ‹φ{y1...yn}› using indisc[THEN "≡E"(1)] "&E" by blast
7437          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7438        }
7439        moreover {
7440          AOT_assume 0: ‹(¬((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn}))) & φ{y1...yn})›
7441          AOT_hence ‹∀x1...∀xn ¬(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})›
7442            using "&E"(1) "cqt-further:4" "→E" by blast
7443          AOT_hence ‹¬(∀F([F]y1...yn ≡ [F]y1...yn) & φ{y1...yn})› using "∀E" by blast
7444          AOT_hence ‹¬∀F([F]y1...yn ≡ [F]y1...yn) ∨ ¬φ{y1...yn}›
7445            using "≡E"(1) "oth-class-taut:5:c" by blast
7446          moreover AOT_have ‹∀F([F]y1...yn ≡ [F]y1...yn)›
7447            by (simp add: "oth-class-taut:3:a" "universal-cor")
7448          ultimately AOT_have ‹¬φ{y1...yn}› by (metis "¬¬I" "∨E"(2))
7449          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7450        }
7451        ultimately AOT_show ‹p & ¬p› for p using "∨E"(3) "raa-cor:1" by blast
7452      qed
7453    }
7454  qed
7455  moreover AOT_assume ‹□∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7456  ultimately AOT_have ‹□∀y1...∀yn((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn})›
7457    using "→E" by blast
7458  AOT_thus ‹[λx1...xn φ{x1...xn}]↓›
7459    by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7460qed
7461
7462AOT_theorem "kirchner-thm-cor:1": ‹[λx φ{x}]↓ → ∀x∀y(∀F([F]x ≡ [F]y) → □(φ{x} ≡ φ{y}))›
7463proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7464  fix x y
7465  AOT_assume ‹[λx φ{x}]↓›
7466  AOT_hence ‹□∀x∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7467    by (rule "kirchner-thm:1"[THEN "≡E"(1)])
7468  AOT_hence ‹∀x□∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7469    using CBF[THEN "→E"] by blast
7470  AOT_hence ‹□∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7471    using "∀E" by blast
7472  AOT_hence ‹∀y □(∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7473    using CBF[THEN "→E"] by blast
7474  AOT_hence ‹□(∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7475    using "∀E" by blast
7476  AOT_hence ‹□∀F ([F]x ≡ [F]y) → □(φ{x} ≡ φ{y})›
7477    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7478  moreover AOT_assume ‹∀F([F]x ≡ [F]y)›
7479  ultimately AOT_show ‹□(φ{x} ≡ φ{y})› using "→E" "ind-nec" by blast
7480qed
7481
7482AOT_theorem "kirchner-thm-cor:2":
7483  ‹[λx1...xn φ{x1...xn}]↓ → ∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → □(φ{x1...xn} ≡ φ{y1...yn}))›
7484proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7485  fix x1xn y1yn
7486  AOT_assume ‹[λx1...xn φ{x1...xn}]↓›
7487  AOT_hence 0: ‹□∀x1...∀xn∀y1...∀yn (∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7488    by (rule "kirchner-thm:2"[THEN "≡E"(1)])
7489  AOT_have ‹∀x1...∀xn∀y1...∀yn □(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7490  proof(rule GEN; rule GEN)
7491    fix x1xn y1yn
7492    AOT_show ‹□(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7493      apply (rule "RM:1"[THEN "→E", rotated, OF 0]; rule "→I")
7494      using "∀E" by blast
7495  qed
7496  AOT_hence ‹∀y1...∀yn □(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7497    using "∀E" by blast
7498  AOT_hence ‹□(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7499    using "∀E" by blast
7500  AOT_hence ‹□(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7501    using "∀E" by blast
7502  AOT_hence 0: ‹□∀F ([F]x1...xn ≡ [F]y1...yn) → □(φ{x1...xn} ≡ φ{y1...yn})›
7503    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7504  moreover AOT_assume ‹∀F([F]x1...xn ≡ [F]y1...yn)›
7505  moreover AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]↓› by "cqt:2[lambda]"
7506  ultimately AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]x1...xn ≡ [λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]y1...yn›
7507    using "∀E" by blast
7508  moreover AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]y1...yn›
7509    apply (rule "β←C"(1))
7510      apply "cqt:2[lambda]"
7511     apply (fact "cqt:2[const_var]"[axiom_inst])
7512    by (simp add: RN GEN "oth-class-taut:3:a")
7513  ultimately AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]x1...xn› using "≡E"(2) by blast
7514  AOT_hence ‹□∀F ([F]x1...xn ≡ [F]y1...yn)›
7515    using "β→C"(1) by blast
7516  AOT_thus ‹□(φ{x1...xn} ≡ φ{y1...yn})› using "→E" 0 by blast
7517qed
7518
7519AOT_define propositional :: ‹Π ⇒ φ› (‹Propositional'(_')›)
7520  "prop-prop1": ‹Propositional([F]) ≡df ∃p(F = [λy p])›
7521
7522AOT_theorem "prop-prop2:1": ‹∀p [λy p]↓›
7523  by (rule GEN) "cqt:2[lambda]"
7524
7525AOT_theorem "prop-prop2:2": ‹[λν φ]↓›
7526  by "cqt:2[lambda]"
7527
7528AOT_theorem "prop-prop2:3": ‹F = [λy p] → □∀x([F]x ≡ p)›
7529proof (rule "→I")
7530  AOT_assume 0: ‹F = [λy p]›
7531  AOT_show ‹□∀x([F]x ≡ p)›
7532    by (rule "rule=E"[rotated, OF 0[symmetric]]; rule RN; rule GEN; rule "beta-C-meta"[THEN "→E"])
7533      "cqt:2[lambda]"
7534qed
7535
7536AOT_theorem "prop-prop2:4": ‹Propositional([F]) → □Propositional([F])›
7537proof(rule "→I")
7538  AOT_assume ‹Propositional([F])›
7539  AOT_hence ‹∃p(F = [λy p])› using "≡dfE"[OF "prop-prop1"] by blast
7540  then AOT_obtain p where ‹F = [λy p]› using "∃E"[rotated] by blast
7541  AOT_hence ‹□(F = [λy p])› using "id-nec:2" "modus-tollens:1" "raa-cor:3" by blast
7542  AOT_hence ‹∃p □(F = [λy p])› using "∃I" by fast
7543  AOT_hence 0: ‹□∃p (F = [λy p])› by (metis Buridan "vdash-properties:10")
7544  AOT_thus ‹□Propositional([F])›
7545    using "prop-prop1"[THEN "≡Df"]
7546    by (AOT_subst ‹Propositional([F])› ‹∃p (F = [λy p])›) auto
7547qed
7548
7549AOT_define indicriminate :: ‹Π ⇒ φ› ("Indiscriminate'(_')")
7550  "prop-indis": ‹Indiscriminate([F]) ≡df F↓ & □(∃x [F]x → ∀x [F]x)›
7551
7552AOT_theorem "prop-in-thm": ‹Propositional([Π]) → Indiscriminate([Π])›
7553proof(rule "→I")
7554  AOT_assume ‹Propositional([Π])›
7555  AOT_hence ‹∃p Π = [λy p]› using "≡dfE"[OF "prop-prop1"] by blast
7556  then AOT_obtain p where Π_def: ‹Π = [λy p]› using "∃E"[rotated] by blast
7557  AOT_show ‹Indiscriminate([Π])›
7558  proof (rule "≡dfI"[OF "prop-indis"]; rule "&I")
7559    AOT_show ‹Π↓›
7560      using Π_def by (meson "t=t-proper:1" "vdash-properties:6")
7561  next
7562    AOT_show ‹□(∃x [Π]x → ∀x [Π]x)›
7563    proof (rule "rule=E"[rotated, OF Π_def[symmetric]]; rule RN; rule "→I"; rule GEN)
7564      AOT_modally_strict {
7565        AOT_assume ‹∃x [λy p]x›
7566        then AOT_obtain a where ‹[λy p]a› using "∃E"[rotated] by blast
7567        AOT_hence 0: ‹p› by (metis "β→C"(1))
7568        AOT_show ‹[λy p]x› for x
7569          apply (rule "β←C"(1))
7570            apply "cqt:2[lambda]"
7571           apply (fact "cqt:2[const_var]"[axiom_inst])
7572          by (fact 0)
7573      }
7574    qed
7575  qed
7576qed
7577
7578AOT_theorem "prop-in-f:1": ‹Necessary([F]) → Indiscriminate([F])›
7579proof (rule "→I")
7580  AOT_assume ‹Necessary([F])›
7581  AOT_hence 0: ‹□∀x1...∀xn [F]x1...xn› using "≡dfE"[OF "contingent-properties:1"] by blast
7582  AOT_show ‹Indiscriminate([F])›
7583    by (rule "≡dfI"[OF "prop-indis"])
7584       (metis "0" "KBasic:1" "&I" "ex:1:a" "rule-ui:2[const_var]" "vdash-properties:6") 
7585qed
7586
7587AOT_theorem "prop-in-f:2": ‹Impossible([F]) → Indiscriminate([F])›
7588proof (rule "→I")
7589  AOT_modally_strict {
7590    AOT_have ‹∀x ¬[F]x → (∃x [F]x → ∀x [F]x)›
7591      by (metis "instantiation" "cqt-orig:3" "Hypothetical Syllogism" "deduction-theorem" "raa-cor:3")
7592  }
7593  AOT_hence 0: ‹□∀x ¬[F]x → □(∃x [F]x → ∀x [F]x)›
7594    by (rule "RM:1")
7595  AOT_assume ‹Impossible([F])›
7596  AOT_hence ‹□∀x ¬[F]x› using "≡dfE"[OF "contingent-properties:2"] "&E" by blast
7597  AOT_hence 1: ‹□(∃x [F]x → ∀x [F]x)› using 0 "→E" by blast
7598  AOT_show ‹Indiscriminate([F])›
7599    by (rule "≡dfI"[OF "prop-indis"]; rule "&I")
7600       (simp add: "ex:1:a" "rule-ui:2[const_var]" 1)+
7601qed
7602
7603AOT_theorem "prop-in-f:3:a": ‹¬Indiscriminate([E!])›
7604proof(rule "raa-cor:2")
7605  AOT_assume ‹Indiscriminate([E!])›
7606  AOT_hence 0: ‹□(∃x [E!]x → ∀x [E!]x)›
7607    using "≡dfE"[OF "prop-indis"] "&E" by blast
7608  AOT_hence ‹◇∃x [E!]x → ◇∀x [E!]x›
7609    using "KBasic:13" "vdash-properties:10" by blast
7610  moreover AOT_have ‹◇∃x [E!]x›
7611    by (simp add: "thm-cont-e:3")
7612  ultimately AOT_have ‹◇∀x [E!]x›
7613    by (metis "vdash-properties:6")
7614  AOT_thus ‹p & ¬p› for p
7615    by (metis "≡dfE" "conventions:5" "o-objects-exist:5" "reductio-aa:1")
7616qed
7617
7618AOT_theorem "prop-in-f:3:b": ‹¬Indiscriminate([E!]-)›
7619proof (rule "rule=E"[rotated, OF "rel-neg-T:2"[symmetric]]; rule "raa-cor:2")
7620  AOT_assume ‹Indiscriminate([λx ¬[E!]x])›
7621  AOT_hence 0: ‹□(∃x [λx ¬[E!]x]x → ∀x [λx ¬[E!]x]x)›
7622    using "≡dfE"[OF "prop-indis"] "&E" by blast
7623  AOT_hence ‹□∃x [λx ¬[E!]x]x → □∀x [λx ¬[E!]x]x›
7624    using "→E" "qml:1" "vdash-properties:1[2]" by blast
7625  moreover AOT_have ‹□∃x [λx ¬[E!]x]x›
7626    apply (AOT_subst ‹[λx ¬E!x]x› ‹¬E!x› for: x)
7627    apply (rule "beta-C-meta"[THEN "→E"])
7628     apply "cqt:2[lambda]"
7629    by (metis (full_types) "B◇" RN "T◇" "cqt-further:2" "o-objects-exist:5" "vdash-properties:10")
7630  ultimately AOT_have 1: ‹□∀x [λx ¬[E!]x]x›
7631    by (metis "vdash-properties:6")
7632  AOT_hence ‹□∀x ¬[E!]x›
7633    by (AOT_subst (reverse) ‹¬[E!]x›  ‹[λx ¬[E!]x]x› for: x)
7634       (auto intro!: "cqt:2" "beta-C-meta"[THEN "→E"])
7635  AOT_hence ‹∀x □¬[E!]x› by (metis "CBF" "vdash-properties:10")
7636  moreover AOT_obtain a where abs_a: ‹O!a›
7637    using "instantiation" "o-objects-exist:1" "qml:2" "vdash-properties:1[2]" "vdash-properties:6" by blast
7638  ultimately AOT_have ‹□¬[E!]a› using "∀E" by blast
7639  AOT_hence 2: ‹¬◇[E!]a› by (metis "≡dfE" "conventions:5" "reductio-aa:1")
7640  AOT_have ‹A!a›
7641    apply (rule "=dfI"(2)[OF AOT_abstract])
7642     apply "cqt:2[lambda]"
7643    apply (rule "β←C"(1))
7644      apply "cqt:2[lambda]"
7645    using "cqt:2[const_var]"[axiom_inst] apply blast
7646    by (fact 2)
7647  AOT_thus ‹p & ¬p› for p using abs_a
7648    by (metis "≡E"(1) "oa-contingent:2" "reductio-aa:1")
7649qed
7650
7651AOT_theorem "prop-in-f:3:c": ‹¬Indiscriminate(O!)›
7652proof(rule "raa-cor:2")
7653  AOT_assume ‹Indiscriminate(O!)›
7654  AOT_hence 0: ‹□(∃x O!x → ∀x O!x)›
7655    using "≡dfE"[OF "prop-indis"] "&E" by blast
7656  AOT_hence ‹□∃x O!x → □∀x O!x›
7657    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7658  moreover AOT_have ‹□∃x O!x›
7659    using "o-objects-exist:1" by blast
7660  ultimately AOT_have ‹□∀x O!x›
7661    by (metis "vdash-properties:6")
7662  AOT_thus ‹p & ¬p› for p
7663    by (metis "o-objects-exist:3" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7664qed
7665
7666AOT_theorem "prop-in-f:3:d": ‹¬Indiscriminate(A!)›
7667proof(rule "raa-cor:2")
7668  AOT_assume ‹Indiscriminate(A!)›
7669  AOT_hence 0: ‹□(∃x A!x → ∀x A!x)›
7670    using "≡dfE"[OF "prop-indis"] "&E" by blast
7671  AOT_hence ‹□∃x A!x → □∀x A!x›
7672    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7673  moreover AOT_have ‹□∃x A!x›
7674    using "o-objects-exist:2" by blast
7675  ultimately AOT_have ‹□∀x A!x›
7676    by (metis "vdash-properties:6")
7677  AOT_thus ‹p & ¬p› for p
7678    by (metis "o-objects-exist:4" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7679qed
7680
7681AOT_theorem "prop-in-f:4:a": ‹¬Propositional(E!)›
7682  using "modus-tollens:1" "prop-in-f:3:a" "prop-in-thm" by blast
7683
7684AOT_theorem "prop-in-f:4:b": ‹¬Propositional(E!-)›
7685  using "modus-tollens:1" "prop-in-f:3:b" "prop-in-thm" by blast
7686
7687AOT_theorem "prop-in-f:4:c": ‹¬Propositional(O!)›
7688  using "modus-tollens:1" "prop-in-f:3:c" "prop-in-thm" by blast
7689
7690AOT_theorem "prop-in-f:4:d": ‹¬Propositional(A!)›
7691  using "modus-tollens:1" "prop-in-f:3:d" "prop-in-thm" by blast
7692
7693AOT_theorem "prop-prop-nec:1": ‹◇∃p (F = [λy p]) → ∃p(F = [λy p])›
7694proof(rule "→I")
7695  AOT_assume ‹◇∃p (F = [λy p])›
7696  AOT_hence ‹∃p ◇(F = [λy p])›
7697    by (metis "BF◇" "vdash-properties:10")
7698  then AOT_obtain p where ‹◇(F = [λy p])› using "∃E"[rotated] by blast
7699  AOT_hence ‹F = [λy p]› by (metis "derived-S5-rules:2" emptyE "id-nec:2" "vdash-properties:6")
7700  AOT_thus ‹∃p(F = [λy p])› by (rule "∃I")
7701qed
7702
7703AOT_theorem "prop-prop-nec:2": ‹∀p (F ≠ [λy p]) → □∀p(F ≠ [λy p])›
7704proof(rule "→I")
7705  AOT_assume ‹∀p (F ≠ [λy p])›
7706  AOT_hence ‹(F ≠ [λy p])› for p
7707    using "∀E" by blast
7708  AOT_hence ‹□(F ≠ [λy p])› for p
7709    by (rule "id-nec2:2"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7710  AOT_hence ‹∀p □(F ≠ [λy p])› by (rule GEN)
7711  AOT_thus ‹□∀p (F ≠ [λy p])› using BF[THEN "→E"] by fast
7712qed
7713
7714AOT_theorem "prop-prop-nec:3": ‹∃p (F = [λy p]) → □∃p(F = [λy p])›
7715proof(rule "→I")
7716  AOT_assume ‹∃p (F = [λy p])›
7717  then AOT_obtain p where ‹(F = [λy p])› using "∃E"[rotated] by blast
7718  AOT_hence ‹□(F = [λy p])› by (metis "id-nec:2" "vdash-properties:6")
7719  AOT_hence ‹∃p□(F = [λy p])› by (rule "∃I")
7720  AOT_thus ‹□∃p(F = [λy p])› by (metis Buridan "vdash-properties:10")
7721qed
7722
7723AOT_theorem "prop-prop-nec:4": ‹◇∀p (F ≠ [λy p]) → ∀p(F ≠ [λy p])›
7724proof(rule "→I")
7725  AOT_assume ‹◇∀p (F ≠ [λy p])›
7726  AOT_hence ‹∀p ◇(F ≠ [λy p])› by (metis "Buridan◇" "vdash-properties:10")
7727  AOT_hence ‹◇(F ≠ [λy p])› for p
7728    using "∀E" by blast
7729  AOT_hence ‹F ≠ [λy p]› for p
7730    by (rule "id-nec2:3"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7731  AOT_thus ‹∀p (F ≠ [λy p])› by (rule GEN)
7732qed
7733
7734AOT_theorem "enc-prop-nec:1": ‹◇∀F (x[F] → ∃p(F = [λy p])) → ∀F(x[F] → ∃p (F = [λy p]))›
7735proof(rule "→I"; rule GEN; rule "→I")
7736  fix F
7737  AOT_assume ‹◇∀F (x[F] → ∃p(F = [λy p]))›
7738  AOT_hence ‹∀F ◇(x[F] → ∃p(F = [λy p]))›
7739    using "Buridan◇" "vdash-properties:10" by blast
7740  AOT_hence 0: ‹◇(x[F] → ∃p(F = [λy p]))› using "∀E" by blast
7741  AOT_assume ‹x[F]›
7742  AOT_hence ‹□x[F]› by (metis "en-eq:2[1]" "≡E"(1))
7743  AOT_hence ‹◇∃p(F = [λy p])›
7744    using 0 by (metis "KBasic2:4" "≡E"(1) "vdash-properties:10")
7745  AOT_thus ‹∃p(F = [λy p])›
7746    using "prop-prop-nec:1"[THEN "→E"] by blast
7747qed
7748
7749AOT_theorem "enc-prop-nec:2": ‹∀F (x[F] → ∃p(F = [λy p])) → □∀F(x[F] → ∃p (F = [λy p]))›
7750  using "derived-S5-rules:1"[where Γ="{}", simplified, OF "enc-prop-nec:1"]
7751  by blast
7752
7753(*<*)
7754end
7755(*>*)